Join and meet: Exercises

Try these exercises to test your knowledge of joins and meets.

Tangled up
--------------------

![a big crazy poset](http://i.imgur.com/cYqkKm7.png)

Determine whether or not the following joins and meets exist in the poset depicted by the above Hasse diagram.

$c ee b$

%%hidden(Show solution):
This join does *not* exist, because $i$ and $j$ are incomparable upper bounds of $\{ c, b \}$, and no smaller upper bounds of $\{ c, b \}$ exist.
%%

$g ee e$

%%hidden(Show solution):
This join exists. $g ee e = j$.
%%

$j \wedge f$

%%hidden(Show solution):
This meet does *not* exist, because $d$ and $b$ are incomparable lower bounds of $\{ j, f \}$, and no larger lower bounds of $\{ j, f \}$ exist.
%%

$igvee \{a,b,c,d,e,f,g,h,i,j,k,l\}$

%%hidden(Show solution):
This join exists. It is $l$.
%%

Join fu
------------

Let $P$ be a poset, $S \subseteq P$, and $p \in P$. Prove that if both $igvee S$ and $(igvee S) ee p$ exist then $igvee (S \cup \{p\})$ exists as well, and $(igvee S) ee p = igvee (S \cup \{p\})$.

%%hidden(Show solution):
For any $X \subset P$, let $X^U$ denote the set of upper bounds of $X$. The above proposition follows from the fact that $\{igvee S, p\}^U = (S \cup p)^U$, which is apparent from the following chain of bi-implications:

$q \in \{igvee S, p\}^U \iff$

for all $s \in S, q \geq igvee S \geq s$, and $q \geq p \iff$

$q \in (S \cup \{p\})^U$.

If two subsets of a poset have the same set of upper bounds, then either they both lack a least upper bound, or both have the same least upper bound.
%%

Meet fu
------------

Let $P$ be a poset, $S \subseteq P$, and $p \in P$. Prove that if both $igwedge S$ and $(igwedge S) \wedge p$ exist then $igwedge (S \cup \{p\})$ exists as well, and $(igwedge S) \wedge p = igwedge(S \cup \{p\})$.

%%hidden(Show solution):
Note that the proposition we are trying to prove here is the dual of the one stated in join fu. Thanks to the duality principle, this theorem therefore comes for free with our solution to join fu.
%%

Quite a big join
--------------------------

In the poset $\langle \mathbb N, |
angle$ discussed in [43s], does $igvee \mathbb N$ exist? If so, what is it?


Stabiliser is a subgroup

[summary: Given a [-3gd group] $G$ [-3t9 acting] on a [-3jz set] $X$, the [-4mz stabiliser] of some [-element_mathematics element] $x \in X$ is a [-subgroup subgroup] of $G$.
]

Let $G$ be a [-3gd] which [3t9 acts] on the [-3jz set] $X$.
Then for every $x \in X$, the [4mz stabiliser] $\mathrm{Stab}_G(x)$ is a [-subgroup subgroup] of $G$.

# Proof

We must check the group axioms.

- The [-54p identity], $e$, is in the stabiliser because $e(x) = x$; this is part of the definition of a group action.
- [-3gy Closure] is satisfied: if $g(x) = x$ and $h(x) = x$, then $(gh)(x) = g(h(x))$ by definition of a group action, but that is $g(x) = x$.
- [-3h4 Associativity] is inherited from the parent group.
- [-inverse_mathematics Inverses]: if $g(x) = x$ then $g^{-1}(x) = g^{-1} g(x) = e(x) = x$.


Prime number

A [-45h] $n > 1$ is *prime* if it has no [divisor_number_theory divisors] other than itself and $1$.
Equivalently, it has the property that if $n \mid ab$ %%note:That is, $n$ divides the product $ab$%% then $n \mid a$ or $n \mid b$.
Conventionally, $1$ is considered to be neither prime nor [composite_number composite] (i.e. non-prime).

# Examples

- The number $2$ is prime, because its divisors are $1$ and $2$; therefore it has no divisors other than itself and $1$.
- The number $3$ is also prime, as are $5, 7, 11, 13, \dots$.
- The number $4$ is not prime; neither are $6, 8, 9, 10, 12, \dots$.

# Properties

- There are infinitely many primes. ([54r Proof.])
- Every natural number may be written as a product of primes; moreover, this can only be done in one way (if we count "the same product but with the order swapped" as being the same: for example, $2 	imes 3 = 3 	imes 2$ is just one way of writing $6$). ([fundamental_theorem_of_arithmetic Proof.])

# How to find primes

If we want to create a list of all the primes below a given number, or the first $n$ primes for some fixed $n$, then an efficient way to do it is the [sieve_of_eratosthenes Sieve of Eratosthenes].
(There are other sieves available, but Eratosthenes is the simplest.)

There are many [primality_testing tests] for primality and for compositeness.

# More general concept

This definition of "prime" is, in a more general [3gq ring-theoretic] setting, known instead as the property of [5m1 irreducibility].
Confusingly, there is a slightly different notion in this ring-theoretic setting, which goes by the name of "prime"; this notion has [prime_element_ring_theory a separate page on Arbital].
In the ring of integers, the two ideas of "prime" and "irreducible" actually coincide, but that is because the integers form a ring with several very convenient properties: in particular, being a [euclidean_domain Euclidean domain], they are a [-principal_ideal_domain] (PID), and [pid_implies_ufd PIDs have unique factorisation].

[todo: add requisite for divisor_number_theory]


Group orbits partition

Let $G$ be a [-3gd], [3t9 acting] on the set $X$.
Then the [group_orbit orbits] of $X$ under $G$ form a [set_partition partition] of $X$.

# Proof

We need to show that every element of $X$ is in an orbit, and that if $x \in X$ lies in two orbits then they are the same orbit.

Certainly $x \in X$ lies in an orbit: it lies in the orbit $\mathrm{Orb}_G(x)$, since $e(x) = x$ where $e$ is the identity of $G$.
(This follows by the definition of an action.)

Suppose $x$ lies in both $\mathrm{Orb}_G(a)$ and $\mathrm{Orb}_G(b)$, where $a, b \in X$.
Then $g(a) = h(b) = x$ for some $g, h \in G$.
This tells us that $h^{-1}g(a) = b$, so in fact $\mathrm{Orb}_G(a) = \mathrm{Orb}_G(b)$; it is an exercise to prove this formally.

%%hidden(Show solution):
Indeed, if $r \in \mathrm{Orb}_G(b)$, then $r = k(b)$, say, some $k \in G$.
Then $r = k(h^{-1}g(a)) = kh^{-1}g(a)$, so $r \in \mathrm{Orb}_G(a)$.

Conversely, if $r \in \mathrm{Orb}_G(a)$, then $r = m(b)$, say, some $m \in G$.
Then $r = m(g^{-1}h(b)) = m g^{-1} h (b)$, so $r \in \mathrm{Orb}_G(b)$.
%%


Product (Category Theory)

This simultaneously captures the concept of a product of [-3jz sets], [-3rb posets], [-3gd groups], [-topological_space topological spaces] etc. In addition, like any [-universal_property universal construction], this characterization does not differentiate between [-4f4 isomorphic] versions of the product, thus allowing one to abstract away from an arbitrary, [-specific_construction_category_theory specific construction].

##Definition##
Given a pair of objects $X$ and $Y$ in a category $\mathbb{C}$, the **product** of $X$ and $Y$ is an object $P$ *along with a pair of morphisms* $f: P
ightarrow X$ and $g: P
ightarrow Y$ satisfying the following [-universal_property universal] condition:

Given any other object $W$ and morphisms $u: W
ightarrow X$ and $v:W
ightarrow Y$ there is a *unique* morphism $h: W
ightarrow P$ such that $fh = u$ and $gh = v$.


Empirical probabilities are not exactly 0 or 1

[summary:  [Cromwell's Rule](https://en.wikipedia.org/wiki/Cromwell%27s_rule) in statistics forbids us to assign probabilities of exactly $0$ or $1$ to any empirical proposition - that is, it is always possible to be mistaken.

- Probabilities of exactly $0$ or $1$ correspond to infinite [1zh log odds], meaning that no finite amount of evidence can ever suffice to reach them, or overturn them.  Once you assign probability $0$ or $1,$ you can never change your mind.
- Sensible [4mr universal priors] never assign probability exactly $0$ or $1$ to any predicted future observation - their hypothesis space is always broad enough to include a scenario where the future is different from the past.]

[Cromwell's Rule](https://en.wikipedia.org/wiki/Cromwell%27s_rule) in statistics argues that no empirical proposition should be assigned a subjective probability of *exactly* $0$ or $1$ - it is always *possible* to be mistaken.  (Some argue that this rule should be generalized to logical facts as well.)

A probability of exactly $0$ or $1$ corresponds to infinite [1zh log odds], and would require infinitely [22x strong] evidence to reach starting from any finite [1rm prior].  To put it another way, if you don't start out infinitely certain of a fact before making any observations (before you were born), you won't reach infinite certainty after any finite number of observations involving finite probabilities.

All sensible [4mr universal priors] seem so far to have the property that they never assign probability exactly $0$ or $1$ to any predicted future observation, since their hypothesis space is always broad enough to include an imaginable state of affairs in which the future is different from the past.

If you did assign a probability of exactly $0$ or $1,$ you would be unable to [1ly update] no matter how much contrary evidence you observed.  [1rb Prior odds] of 0 : 1 (or 1 : 0), times any finite [1rq likelihood ratio], end up yielding 0 : 1 (or 1 : 0).

As Rafal Smigrodski put it:

> "I am not totally sure I have to be always unsure.  Maybe I could be legitimately sure about something.  But once I assign a probability of 1 to a proposition, I can never undo it.  No matter what I see or learn, I have to reject everything that disagrees with the axiom.  I don't like the idea of not being able to change my mind, ever."


Universal prior

A "universal" [27p prior] is a [3tb probability distribution] that assigns positive probability to *every thinkable hypothesis*, for some reasonable meaning of "every thinkable hypothesis".  A central example would be [11w Solomonoff induction], in which the observations are a sequence of bits, the universal prior is "every possible computer program that generates bit sequences", and every such computer program starts with a positive probability.


Stabiliser (of a group action)

[summary: The stabilizer of an element $x$ under the action of a group $G$ is the set ([4lt actually a subgroup]) of elements of $G$ which leave $x$ unchanged. ]

Let the [-3gd] $G$ [3t9 act] on the set $X$.
Then for each element $x \in X$, the *stabiliser* of $x$ under $G$ is $\mathrm{Stab}_G(x) = \{ g \in G: g(x) = x \}$.
That is, it is the collection of elements of $G$ which do not move $x$ under the action.

The stabiliser of $x$ is a subgroup of $G$, for any $x \in X$. ([4lt Proof.])

A closely related notion is that of the [4v8 orbit] of $x$, and the very important [4l8 Orbit-Stabiliser theorem] linking the two.


Exercise

One way readers can reinforce or make sure they've understood a concept is to do exercises. These pages give a collection of questions for this purpose.


Why is the decimal expansion of log2(3) infinite?

[summary:
It takes more than one but less than two [binary_digit binary digits] to encode a [4sj 3-digit], so $\log_2(3)$ must be between 1 and 2. ([427 Wait, what?]). It takes more than 15 but less than 16 binary digits to encode ten 3-digits, so $10 \cdot \log_2(3)$ must be between 15 and 16, which  means $1.5 < \log_2(3) < 1.6.$ It takes more than 158 but less than 159 binary digits to encode a hundred 3-digits, so $1.58 < \log_2(3) < 1.59.$ And so on. Because no power of 3 is ever equal to any power of 2, $10^n \cdot \log_2(3)$ will never quite be a whole number, no matter how large $n$ is.]

$\log_2(3)$ starts with

1.5849625007211561814537389439478165087598144076924810604557526545410982277943585625222804749180882420909806624750591673437175524410609248221420839506216982994936575922385852344415825363027476853069780516875995544737266834624612364248850047581810676961316404807130823233281262445248670633898014837234235783662478390118977006466312634223363341821270106098049177472541357330110499026268818251703576994712157113638912494135752192998699040767081539505404488360

and goes on indefinitely. Why is it 1.58... in particular? Well, it takes more than one but less than two [binary_digit binary digits] to encode a [4sj 3-digit], so $\log_2(3)$ must be between 1 and 2. ([427 Wait, what?]). It takes more than 15 but less than 16 binary digits to encode ten 3-digits, so $10 \cdot \log_2(3)$ must be between 15 and 16, which  means $1.5 < \log_2(3) < 1.6.$ It takes more than 158 but less than 159 binary digits to encode a hundred 3-digits, so $1.58 < \log_2(3) < 1.59.$ And so on. Because no power of 3 is ever equal to any power of 2, $10^n \cdot \log_2(3)$ will never quite be a whole number, no matter how large $n$ is.

Thus, $\log_2(3)$ has no finite decimal expansion, because $3$ is not a [4zq rational] [-power] of $2$. Using this argument, we can see that $\log_b(x)$ is an integer if (and only if) $x$ is a power of $b$, and that $\log_b(x)$ only has a finite expansion if some power of $x$ is a power of $b.$


Guide

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Needs brief summary

Pages which have a long or in-depth main summary should also have a brief summary. These show up as an alternate tab on the [1kl summary popover] when a user hovers over a [17f greenlink]. Add this tag to pages that need to have a brief summary.

Pages may also require a [433 general] or [4pt technical] summary.


n-digit

An $n$-digit is a physical object that can be stably placed into any of $n$ distinguishable states. For example, a coin (which can be placed heads or tails) and a single bit of memory on a computer (which either has a high volt level or a low volt level) are both examples of 2-digits. A [-42d] is an example of a 10-digit. One die is an example of a 6-digit; two dice together are an example of a 36-digit (because they can be placed in 36 different ways).

What does and doesn't count as an $n$-digit depends on context and convention: For example, if you want to communicate a message to me by placing a penny heads-side up and choosing whether to point Abraham Lincoln's face either north, south, east, or west, then, for the purposes of the two of us, that penny is a 4-digit rather than a 2-digit. The definition of "stably placed" is also a bit up-for-grabs: If you're writing a computer program and need to store a [3v9 256-message] in short-term memory, then a byte of RAM will do, but if you need to store the same 256-message for a long period of time, you may need to use a less temporary 256-digit (such as a hard drive).

Note that it's possible to emulate $m$-digits using $n$-digits, in general. If $m < n$ then an $n$-digit is trivially an $m$-digit (i.e., you can use a digit wheel like a 7-digit in a pinch), and if $m > n$ then, given enough $n$-digits, you can make do. For example, 3 coins can be used to encode an 8-digit. See also [emulating_digits].


Emulating digits

In general, given enough $n$-digits, you can emulate an $m$-digit, for any $m, n \in$ [45h $\mathbb N$]. If $m < n,$ you can emulate an $m$-digit using just one $n$-digit — in other words, you can use a [-42d] like a $7$-digit if you want to, by just ignoring three of the possible ways to set the digit wheel. If $m > n,$ things are a bit more difficult, but only slightly.

Basically, with 2 $n$-digits, you can emulate a $n^2$-digit, as follows. Using your two $n$-digits, encode a number $(x, y)$ where $0 \le x < n$ and $0 \le y < n$. Interpret $(x, y)$ as $xn + y.$ You have now encoded a number between 0 (if  $x = y = 0$) and $n^2 - 1$ (if $x = y = n-1$). Congratulations, you just used two $n$-digits to make an $n^2$ digit!

You can use the same strategy to emulate $n^3$-digits (interpret $(x, y, z)$ as $xn^2 + yn + z$), $n^4$-digits (you get the picture), and so on. Now, to emulate an $m$-digit, just pick an exponent $a$ such that $n^a > m,$ collect $a$ copies of an $n$-digit, and you're done.

This isn't necessarily the most efficient way to use $n$-digits to encode $m$-digits. For example, if $m$ is 1,000,001 and $n$ is 10, then you need seven 10-digits. Seven 10-digits are enough to emulate a 10-million-digit, whereas $m$ is a mere million-and-one-digit — paying for a 10-million-digit when all you needed was an $m$-digit seems a bit excessive. For some different methods you can use to recover your losses when encoding one type of digit using another type of digit, see [44l] and [3ty]. (These techniques are fairly useful in practice, given that modern computers encode everything using [3p0 bits], i.e. 2-digits, and so it's useful to know how to efficiently encode $m$-messages using bits when $m$ is pretty far from the nearest power of 2.)


Decimal notation

Seventeen is the number that represents as many things as there are x marks at the end of this sentence: xxxxxxxxxxxxxxxxx. Writing out numbers by saying "the number representing how many things there are in this pile:" gets unwieldy when the pile gets large. Thus, we _represent_ numbers using the [numeral numerals] 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Specifically, we write the number representing this many things: xxx as "3", and the number representing this many things: xxxxxxxxxxx as 11, and the number seventeen as "17". This is called "decimal notation,"  because there are ten different symbols that we use. Numbers don't have to be written down in decimal notation, it's also possible to write them down in other notations such as [-binary_notation]. Some numbers can't even be written out in decimal notation (in full); consider, for example, the number $e$ which, in decimal notation, starts out with the digits 2.71828... and just keeps going.

# How decimal notation works

How do you know that 17 is the number that represents the number of xs in this sequence: xxxxxxxxxxxxxxxxx? In practice, you know this because the rules of decimal notation were ingrained in you in a young child. But do you know those rules explicitly? Could you write out a series of rules for taking in some input symbols like '2', '4', and '6' and using those to figure out how many pebbles to add to a pile?

The answer, of course, is this many:

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

But how do we perform that conversion in general?

In short, the number 246 represents $(2 \cdot 100) + (4 \cdot 10) + (6 \cdot 1),$ so as long as we know how to do addition and multiplication, and as long as we know what the basic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 mean, and as long as we know how to get to [power powers] of 10 (1, 10, 100, 1000, ...), then we can explicitly understand decimal notation.

(What do the basic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 mean? By convention, they represent as many things as are in the following ten sequences of xs: , x, xx, xxx, xxxx, xxxxx, xxxxxx, xxxxxxx, xxxxxxxx, and xxxxxxxxx, respectively.)

This explanation assumes that you're already quite familiar with decimal notation. Explaining decimal notation from scratch to someone who doesn't already know it (which was a task people _actually had to do_ back when half the world was using Roman numerals, a much less convenient system for representing numbers) is a fun task; to see what that looks like, refer to [+representing_numbers_from_scratch].

# Other common notations

The above text made use of [-unary_notation], which is a method of representing numbers by making a number of marks that correspond to the represented number. For example, in unary notation, 17 is written xxxxxxxxxxxxxxxxx (or ||||||||||||||||| or whatever, the actual marks don't matter). This is perhaps somewhat easier to understand, but writing large numbers like 93846793284756 gets rather ungainly.

Historical notations include [https://en.wikipedia.org/wiki/Roman_numerals Roman numerals], which were a pretty bad way to represent numbers. (It took humanity quite some time to find _good_ tools for representing numbers; the decimal notation that's been ingrained in your head since early childhood is the result of many centuries worth of effort. It's much harder to invent good representations of numbers when you don't even have good tools for writing down and reasoning about numbers. Furthermore, the modern tools for representing numbers aren't necessarily ideal!)

Common notations in modern times (aside from decimal notation) include [-binary_notation] (often used by computers), [-hexadecimal_notation] (which is a useful format for humans reading binary notation). Binary notation and hexadecimal notation are very similar to decimal notation, with the difference that binary uses only two distinct symbols (instead of ten), and hexadecimal uses sixteen.


Inverse function

If a function $g$ is the inverse of a function $f$, then $g$ undoes $f$, and $f$ undoes $g$. In other words, $g(f(x)) = x$ and $f(g(y)) = y$. An inverse function takes as its [3js domain] the [3lm range] of the original function, and the [3lm range] of the inverse function is the [3js domain] of the original. To put that another way, if $f$ maps $A$ onto $B$, then $g$ maps $B$ back onto $A$. To indicate the inverse of a function $f$, we write $f^{-1}$.

## Examples ##

$$y=f(x) = x^3\ \ \ \ \ \ \ \ \ \ \ \   x=f^{-1}(y) = y^{1/3}$$
$$y=f(x) = e^x\ \ \ \ \ \ \ \ \ \ \ \   x=f^{-1}(y) = ln(y)$$
$$y=f(x) = x+4\ \ \ \ \ \ \ \ \ \ \ \  x=f^{-1}(y) = y-4$$


Quotient group

[summary:
Given a [-3gd group] $G$ with operation $\bullet$ and a special kind of [-subgroup subgroup] $N \leq G$ called the "[-4h6 normal subgroup]", there is a way of "dividing" $G$ by $N$, written $G/N$. Usually this is defined so that each [-element_mathematics element] of $G/N$ is a [-subset subset] of $G$. In fact, each of them is an [-equivalence_class equivalence class] of elements in $G$, so called because we think of everything in one equivalence class as being equivalent. One of the equivalence classes is $N$, and each of the other equivalence classes can be seen as $N$ "shifted around" inside $G$. Because they are equivalence classes, each element of $G$ only occurs in exactly one of them.

This collection $G/N$ of equivalence classes has a [-group_structure group structure] based on the structure on $G$. Write the [-group_operation operation] as $\circ$. Say $A$ and $B$ are two equivalence classes in $G/N$. Then $A \circ B$ is defined by taking an element $a \in A$ and $b \in B$, multiplying them as $a \bullet b$ and checking in which equivalence class this product ends up.

There are some things to check, namely that the equivalence class in which we end up is the same regardless of which elements in $A$ and $B$ we happen to pick. This is guaranteed by the fact that $N$ is normal in $G$.

There is also a [-47t group homomorphism] $\phi$ from $G$ to $G/N$ that "collapses" the elements by sending each element in $G$ to the equivalence class in which it lies. This is called collapsing because lots of elements in $G$ are sent to the same element in $G/N$, so we can see $G/N$ as a collapsed version of $G$.
]

[summary(brief):
A **quotient group** $G/N$ of a [-3gd group] $G$ by a [-4h6 normal subgroup] $N$ is obtained by dividing up the group into pieces ([-equivalence_class equivalence classes]), and then treating everything in one class the same way (by treating each class as a single element). The quotient group has a group structure defined on it based on the original structure of $G$, that works 'basically the same as $G$ up to equivalence'.
 ]

[summary(technical):
Given a [-3gd group] $(G, \bullet)$ and a [-4h6 normal subgroup] $N \unlhd G$. The **quotient** of $G$ by $N$, written $G/N$, has as [-3gz underlying set] the set of (left)-[-4j4 cosets] of $N$ in $G$ and as operation $\circ$ which is defined as $aN \circ  bN = (a \bullet b) N$, where $xN = \{xn : n \in N\}$ for each $x \in G$.

The operation $\circ$ is [well_defined well defined] in the sense that if other representatives $a'$ and $b'$ are chosen such that
$a'N = aN$ and $b'N = bN$ then also $(a' \bullet b')N = (a \bullet b)N$.

There is a [-canonical canonical] [-47t homomorphism] (sometimes called the [-quotient_projection projection]) $\phi: G \rightarrow G/N: a \mapsto aN$.

This is a special case of a [-quotient_universal_algebra quotient] from universal algebra.
]

[summary(examples):
Given the [-3gd group] of [-48l integers] $\mathbb{Z}$, and the [-nomral_subgroup normal subgroup] $2 \mathbb{Z}$ of all even numbers, we can form the group $\mathbb{Z}/2\mathbb{Z}$. This group has only two [-element elements] and [-personification_in_mathematics only cares] if a number is odd or even. It tells us that the sum of an odd and an even number is odd, that the sum of two even numbers is even, and the sum of two odd numbers is also even!
]

[summary(motivation):
Let's say we have a group. Maybe the group is kinda large and unwieldy, and we want to find an easier way to think about it. Or maybe we just want to focus on a certain aspect of the group. Some of the actions will change things in ways we just don't really care about, or don't mind ignoring for now. So let's create a group homomorphism that will map all these actions to the identity action in a new group. The image of this homomorphism will be a group much like the first, except that it will ignore all the effects that come from those actions that we're ignoring - just what we wanted! This new group is called the quotient group.
]

#The basic idea
Let's say we have a [3gd group]. Maybe the group is kinda large and unwieldy, and we want to find an easier way to think about it. Or maybe we just want to focus on a certain aspect of the group. Some of the actions will change things in ways we just don't really care about, or don't mind ignoring for now. So let's create a [47t group homomorphism] that will map all these actions to the identity action in a new group. The image of this homomorphism will be a group much like the first, except that it will ignore all the effects that come from those actions that we're ignoring - just what we wanted! This new group is called the quotient group.

#Definition

We start with our group $G$. The actions we want to ignore form a group $N$, which must be a [4h6 normal subgroup] of $G$. The quotient group is then called $G/N$, and has a canonical homomorphism $\phi: G \rightarrow G/N$ which maps $g \in G$ to the [4j4 coset] $gN$.

## The divisor group
In the definition, we require the divisor $N$, to be a normal subgroup of $G$. Why? Well first, let's see why requiring $N$ to be a group makes sense. Remember that $N$ has the actions whose effects we want to ignore. So it makes sense that it should contain the identity action, which has no effect. It also is reasonable that it would be closed under the group operation - doing two things we don't care about shouldn't change anything we care about. Together, these two properties imply it is a [subgroup subgroup]: $N \le G$.

A subgroup is great, but it isn't quite good enough by itself to work here. That's because we want the quotient group to preserve the overall structure of the group, i.e. it should preserve the group multiplication. In other words, there needs to be a [47t group homomorphism] $\phi$ from $G$ to $G/N$. Since $N$ is the subgroup of things we want to ignore, all its actions should get mapped to the identity action under this homomorphism. That means it's the [49y kernel] of the homomorphism $\phi$, which means it's a [4h6 normal subgroup]: $N \trianglelefteq G$.

## Cosets
What exactly are the elements of the new group? They are [equivalence_class equivalence classes] of actions, the sets $gN = \{gn : n \in N\}$ where $g \in G$, also known as a [4j4 coset]. The identity element is the set $N$ itself. Multiplication is defined by $g_1N \cdot g_2N = (g_1g_2)N$.

# Generalizes the idea of a quotient
What gives a quotient group the right to call itself a quotient? If $G$ and $N$ both have finite order, then $|G/N| = |G|/|N|$, which can be proved by the fact that $G/N$ consists of the cosets of $N$ in $G$, and that these [4j8 cosets are the same size], and [4j5 partition] $G$.

#Example
Suppose you have a collection of objects, and you need to split them into two equal groups. So you are trying to determine under what circumstances changing the number of objects will affect this property. You notice that changing the size of the collection by certain numbers such as 0, 2, 4, 24, and -6 doesn't affect this property.

The set of different size changes can be modeled as the additive group of integers $\mathbb Z$. The changes that don't affect this property also form a group: $2\mathbb Z = \{2n : n\in \mathbb Z\}$. Exercise: verify that this is a normal subgroup of $\mathbb Z$.

This subgroup gives us two cosets: $0 + 2\mathbb Z$ and $1 + 2\mathbb Z$ (remember that $+$ is the group operation in this example), which are the elements of our quotient group. We will give them their conventional names: $\text{even}$ and $\text{odd}$, and we can apply the coset multiplication rule to see that $\text{even}+ \text{even} = \text{even}$, $\text{even} + \text{odd} = \text{odd}$, and $\text{odd} + \text{odd} = \text{even}$.

Instead of thinking about specific numbers, and how they will change our ability to split our collection of objects into two equal groups, we now have reduced the problem to its essence. Only the parity matters, and it follows the simple rules of the quotient group we discovered.

#See also

 - [4jn Lagrange's theorem].
 - [first_isomorphism_theorem The first isomorphism theorem].


Exponential

[summary: The exponent base $b$ of a number $x$, written $b^x,$ is what you get when you multiply 1 by $b$, $x$ times. For example, the exponent base 10 of 3 is $10^3$ and pronounced "10 to the power 3" is 1000, because $10 \cdot 10 \cdot 10 = 1000$. Similarly, $2^4=16,$ because $2 \cdot 2 \cdot 2 \cdot 2 = 16.$

The input $x$ may be fractional. For example, $10^{1/2}$ is the number you get when you multiply 1 by 10 half a time, where "multiplying by 10 half a time" means multiplying by a number $n$ such that if you multiplied by $n$ twice, you would have multiplied by 10 once. In this case, $n pprox 3.16,$ because then $n \cdot n pprox 10.$]

[todo: Add numerous other summaries. See, e.g., the summaries at /p/logarithm.]
[todo: I suggest making the main page for exponential mirror the main page at /p/logarithm.]

An exponential is a [-3jy] that can be represented by some [-constant] taken to the [power_mathematics power] of a [-variable]. The name comes from the fact that the variable is the *exponent* of the expression.

## Exponential Growth

Exponentials are most useful in describing growth patterns where the growth rate is [4w3 proportional] to the amount of the thing that's growing. They can be represented by the formula: $f(x) = c 	imes a^x$, where $c$ is the starting value and $a$ is the growth factor.

The classic example of exponential growth is [-compound_interest]. If you have \$100 in a bank account that gives you 2% interest every year, then every year your money is multiplied by $1.02$. This means you can represent your account balance as $f(x) = 100 	imes 1.02^x$, where $x$ is the number of years your money has been in the account.

Another example is a dividing cell. If one cell is placed into an infinite culture and splits once every hour, the number of cells in the culture after $x$ hours is $f(x) = 1 	imes 2^x$ (assuming none of the cells die).

### Recursive definition

We mentioned earlier that in an exponential function, the growth rate is proportional to the amount of the thing that's growing. In the compound interest example, we can write each value in terms of the previous value: $f(x) = f(x-1) 	imes 1.02$. Therefore, the amount of growth at every step can be represented as $\Delta f(x) = f(x+1) - f(x) = 0.02 	imes f(x)$.

This makes exponential growth a *memoryless growth function*, as the growth rate depends only on current information. Compare this to [-simple_interest], where the interest only grows at a percentage of the initial value. Then we would have to write: $f(x) = f(x-1) + 0.02 	imes f(0)$, and because $f(0)$ is constant while $f(x)$ continues to grow, we cannot express the growth rate in terms of only current information — we have to keep "in memory" the initial balance to calculate the growth rate.


Most complex things are not very compressible

Although the [46h halting problem] means we can't *prove* it doesn't happen, it would nonetheless be *extremely surprising* if some 100-state Turing machine turned out to print the exact text of Shakespeare's *Romeo and Juliet.*  Unless something was specifically generated by a simple algorithm, the Vast supermajority of data structures that *look* like they have high [5v algorithmic complexity] actually *do* have high algorithmic complexity.  Since there are at most $2^{101}$ programs that can be specified with at most 100 bits (in any particular language), we can't fit all the 1000-bit data structures into all the 100-bit programs.  While *Romeo and Juliet* is certainly highly compressible, relative to most random bitstrings of the same length, it would be shocking for it to compress *all the way down* to a 100-state Turing machine.  There just aren't enough 100-state Turing machines for one of their outputs to be *Romeo and Juliet*.  Similarly, if you start with a 10 kilobyte text file, and 7zip compresses it down to 2 kilobytes, no amount of time spent trying to further compress the file using other compression programs will ever get that file down to 1 byte.  For any given compressor there's at most 256 starting files that can ever be compressed down to 1 byte, and your 10-kilobyte text file almost certainly isn't one of them.

This takes on defensive importance with respect to refuting the probability-theoretic fallacy, "Oh, sure, Occam's Razor seems to say that this proposition is complicated.  But how can you be sure that this apparently complex proposition wouldn't turn out to be generated by some very simple mechanism?"  If we consider a [1rd partition] of 10,000 possible propositions, collectively having a 0.01% probability on average, then all the arguments in the world for why various propositions might have unexpectedly high probability, must still add up to an average probability of 0.01%.  It can't be the case that after considering that proposition 1 might have secretly high probability, and considering that proposition 2 might have secretly high probability, and so on, we end up assigning 5% probability to each proposition, because that would be a total probability of 500.  If we assign prior probabilities using an algorithmic-complexity Occam prior as in [11w Solomonoff induction], then the observation that "most apparently complex things can't be further compressed into an amazingly simple Turing machine", is the same observation as that "most apparently Occam-penalized propositions can't turn out to be simpler than they look" or "most apparently subjectively improbable things can't turn out to have unseen clever arguments that would validly make them more subjectively probable".


Group orbit

When we have a [3gd group] [3t9 acting] on a set, we are often interested in how the group acts on a particular [group_element element]. One natural way to do this is to look at the set of all the elements that different group actions will take the starting element to. This is called the orbit.

#Definition
Let $X$ be a set, with element $x \in X$, and let $G$ be a group acting on $X$. Then the orbit of $x$ is $Gx = \{gx : g \in G\}$.

#Properties
The set $X$ is partitioned by group orbits - each element defines its own orbit, and two orbits containing the same element must be the same, because every action has an inverse action. This gives the fact that [4j5 cosets partition a group] as the special case where $X$ is a subgroup of $G$.
#See Also

 - [4l8 The orbit-stabiliser theorem].


Lagrange theorem on subgroup size: Intuitive version

[summary: Lagrange's Theorem gives us a restriction on the size of a subgroup of a finite group: namely, subgroups have size dividing the parent group.]

Given a finite [-3gd] $G$, it may have many [576 subgroups].
So far, we know almost nothing about those subgroups; it would be great if we had some way of restricting them.

An example of such a restriction, which we do already know, is that a subgroup $H$ of $G$ has to have [3gg size] less than or equal to the size of $G$ itself.
This is because $H$ is contained in $G$, and if the set $X$ is contained in the set $Y$ then the size of $X$ is less than or equal to the size of $Y$.
(This would have to be true for any reasonable definition of "size"; the [4w5 usual definition] certainly has this property.)

Lagrange's Theorem gives us a much more powerful restriction: not only is the size $|H|$ of $H$ less than or equal to $|G|$, but in fact $|H|$ divides $|G|$.

%%hidden(Example: subgroups of the cyclic group on six elements):
*A priori*, all we know about the subgroups of the [-47y] $C_6$ of order $6$ is that they are of order $1, 2, 3, 4, 5$ or $6$.

Lagrange's Theorem tells us that they can only be of order $1, 2, 3$ or $6$: there are no subgroups of order $4$ or $5$.
Lagrange tells us nothing about whether there *are* subgroups of size $1,2,3$ or $6$: only that if we are given a subgroup, then it is of one of those sizes.

In fact, as an aside, there are indeed subgroups of sizes $1,2,3,6$:

- the subgroup containing only the identity is of order $1$
- the "improper" subgroup $C_6$ is of order $6$
- subgroups of size $2$ and $3$ are guaranteed by [4l6 Cauchy's theorem].

%%

# Proof

In order to show that $|H|$ divides $|G|$, we would be done if we could divide the elements of $G$ up into separate buckets of size $|H|$.

There is a fairly obvious place to start: we already have one bucket of size $|H|$, namely $H$ itself (which consists of some elements of $G$).
Can we perhaps use this to create more buckets of size $|H|$?

For motivation: if we think of $H$ as being a collection of symmetries (which we can do, by [49b Cayley's Theorem] which states that all groups may be viewed as collections of symmetries), then we can create more symmetries by "tacking on elements of $G$".

Formally, let $g$ be an element of $G$, and consider $gH = \{ g h : h \in H \}$.

Exercise: every element of $G$ does have one of these buckets $gH$ in which it lies.
%%hidden(Show solution):
The element $g$ of $G$ is contained in the bucket $gH$, because the identity $e$ is contained in $H$ and so $ge$ is in $gH$; but $ge = g$.
%%

Exercise: $gH$ is a set of size $|H|$. %%note:More formally put, [-4j8].%%
%%hidden(Show solution):
In order to show that $gH$ has size $|H|$, it is enough to match up the elements of $gH$ [499 bijectively] with the elements of $|H|$.

We can do this with the [-3jy] $H 	o gH$ taking $h \in H$ and producing $gh$.
This has an [4sn inverse]: the function $gH 	o H$ which is given by pre-multiplying by $g^{-1}$, so that $gx \mapsto g^{-1} g x = x$.
%%

Now, are all these buckets separate? Do any of them overlap?

Exercise: if $x \in rH$ and $x \in sH$ then $rH = sH$. That is, if any two buckets intersect then they are the same bucket. %%note:More formally put, [4j5].%%
%%hidden(Show solution):
Suppose $x \in rH$ and $x \in sH$.

Then $x = r h_1$ and $x = s h_2$, some $h_1, h_2 \in H$.

That is, $r h_1 = s h_2$, so $s^{-1} r h_1 = h_2$.
So $s^{-1} r = h_2 h_1^{-1}$, so $s^{-1} r$ is in $H$ by closure of $H$.

By taking inverses, $r^{-1} s$ is in $H$.

But that means $\{ s h : h \in H \}$ and $\{ r h : h \in H\}$ are equal.
Indeed, we show that each is contained in the other.

- if $a$ is in the right-hand side, then $a = rh$ for some $h$. Then $s^{-1} a = s^{-1} r h$; but $s^{-1} r$ is in $H$, so $s^{-1} r h$ is in $H$, and so $s^{-1} a$ is in $H$.
Therefore $a \in s H$, so $a$ is in the left-hand side.
- if $a$ is in the left-hand side, then $a = sh$ for some $h$. Then $r^{-1} a = r^{-1} s h$; but $r^{-1} s$ is in $H$, so $r^{-1} s h$ is in $H$, and so $r^{-1} a$ is in $H$.
Therefore $a \in rH$, so $a$ is in the right-hand side.
%%

We have shown that the "[4j4 cosets]" $gH$ are all completely disjoint and are all the same size, and that every element lies in a bucket; this completes the proof.


Proportion

A **proportion** is a representation of one value as a [-fraction] or [-multiple] of another value. It is used to provide a sense of relative magnitude.

If some value $a$ is proportional to another value $b$, it means that $a$ can be expressed as $c 	imes b$, where $c$ is some constant, also known as the *constant of proportionality* (or sometimes just the *proportion*) of $a$ to $b$.

For example, no matter how large a domino is, it is always twice as long as it is wide. Then, the proportion of the length compared to the width can be said to be $2$, and we can write $l = 2w$ where $w$ is the width and $l$ is the length.

## Notation

To write that $a$ is proportional to $b$, the notation $a \propto b$ is used. For example, the surface area of an object of a specific shape is proportional to the square of its length, so we write $A \propto L^2$ where $A$ is area and $L$ is length.

## Percentages

Percentages are a way of describing proportions relative to the number 100 to make them easier to understand. When people refer to a "percentage basis" for calculating something, they are specifically talking about this type of proportion.

To get a percentage from a proportion, simply multiply it by 100. In our domino example, we multiply $2$ by $100$ to find that the length is equal to $200\%$ of the width.

Percentages only work with dimensionless proportions — that is, where the constant of proportionality has no [unit units]. For example, if a car requires $6.7$ litres of gas to travel $100$ kilometres, it makes no sense to say that the fuel efficiency of the car (which is a constant of proportionality for fuel used versus distance travelled) is $6.7\%$ litres per kilometre. However, when comparing the fuel efficiencies of two cars, if one car has an efficiency of $5.2$ L/100km and another has one of $6.8$ L/100km, it makes sense to say that the first car is $131\%$ as efficient as the second.

## Common examples of proportions

The [-circumference] of a [-circle] is proportional to its [-diameter]. We write $C \propto d$, and the constant of proportionality is $\pi = 3.<PHONE>\ldots$.

The [-area] of a circle is proportional to the square of its radius. We write $A \propto r^2$, and the constant of proportionality is also $\pi$.

The rate of growth of an [-4ts] function is proportional to the current value of the function. We write $rac{df}{dt} \propto f$, or $\Delta f \propto f$ if considering discrete values.

The gravitational potential energy of an object in a constant gravitational field is proportional to its height relative to some zero point. We write $E \propto h$, and the constant of proportionality is the weight of the object, its mass multiplied by the gravitational acceleration in the field.

The amount of current flowing through a specific wire is proportional to the potential difference, or voltage, between the endpoints of the wire. We write $I \propto V$, and the constant of proportionality is the resistance of that wire.

In an ideal gas, the pressure of the gas is proportional to its temperature, if the volume and number of particles in the gas is held constant. This is one property of the ideal gas law in physics. We write $P \propto T$.

[todo: proportions as ratios]


Cardinality

[summary(brief): The **cardinality** of a [-3jz] is a formalization of the "number of [5xy elements]" in the set.]

[summary: If $A$ is a finite set then the cardinality of $A$, denoted $|A|$, is the number of elements $A$ contains. When $|A| = n$, we say that $A$ is a set of cardinality $n$. There exists a [499 bijection] from any finite set of cardinality $n$ to the set $\{0, ..., (n-1)\}$ containing the first $n$ natural numbers.

Two infinite sets have the same cardinality if there exists a bijection between them. Any set in bijective correspondence with [45h $\mathbb N$] is called __countably infinite__, while any infinite set that is not in bijective correspondence with $\mathbb N$ is call **[2w0 uncountably infinite]**. All countably infinite sets have the same cardinality, whereas there are multiple distinct uncountably infinite cardinalities.]

[summary(technical): The cardinality (or size) $|X|$ of a set $X$ is the number of elements in $X.$ For example, letting $X = \{a, b, c, d\}, |X|=4.$

[todo: technical summary of infinite cardinality]]

The **cardinality** of a [-3jz] is a formalization of the "number of elements" in the set.

Set cardinality is an [-53y]. Two sets have the same cardinality if (and only if) there exists a [499 bijection] between them.

## Definition of equivalence classes

### Finite sets

A set $S$ has a cardinality of a [-45h] $n$ if there exists a bijection between $S$ and the set of natural numbers from $1$ to $n$. For example, the set $\{9, 15, 12, 20\}$ has a bijection with $\{1, 2, 3, 4\}$, which is simply mapping the $m$th element in the first set to $m$; therefore it has a cardinality of $4$.

We can see that this equivalence class is [ well-defined] — if there exist two sets $S$ and $T$, and there exist bijective functions $f : S 	o \{1, 2, 3, \ldots, n\}$ and $g : \{1, 2, 3, \ldots, n\} 	o T$, then $g \circ f$ is a bijection between $S$ and $T$, and so the two sets also have the same cardinality as each other, which is $n$.

The cardinality of a finite set is always a natural number, never a fraction or decimal.

### Infinite sets

Assuming the [69b axiom of choice], the cardinalities of infinite sets are represented by the [aleph_numbers]. A set has a cardinality of $leph_0$ if there exists a bijection between that set and the set of *all* natural numbers. This particular class of sets is also called the class of [-countably_infinite_sets].

Larger infinities (which are [2w0 uncountable]) are represented by higher Aleph numbers, which are $leph_1, leph_2, leph_3,$ and so on through the [ordinal ordinals].

**In the absence of the Axiom of Choice**

Without the axiom of choice, not every set may be [55r well-ordered], so not every set bijects with an [-ordinal], and so not every set bijects with an aleph.
Instead, we may use the rather cunning [Scott_trick].

%%todo: Examples and exercises (possibly as lenses) %%

%%todo: Split off a more accessible cardinality page that explains the difference between finite, countably infinite, and uncountably infinite cardinalities without mentioning alephs, ordinals, or the axiom of choice.%%


Needs clickbait

[summary: This page does not have [597 clickbait]. Clickbait style guidelines:

Yes: "First letter is capital."
Yes: "Pique user's interest in the topic."
Yes: "Most clickbaits are just one sentence telling user what they might find out if they read the page."
Yes: "Can a clickbait ask the reader a leading question?"
Yes: "Period or other punctuation at the end."
Yes: Omitting clickbait if the title is descriptive enough.
No: "Capitalize Every Word"
No: "**Markdown**"]

A meta tag for pages which don't have [597 clickbait]. Clickbait style guidelines:

Yes: "First letter is capital."
Yes: "Pique user's interest in the topic."
Yes: "Most clickbaits are just one sentence telling user what they might find out if they read the page."
Yes: "Can a clickbait ask the reader a leading question?"
Yes: "Period or other punctuation at the end."
Yes: Omitting clickbait if the title is descriptive enough.
No: "Capitalize Every Word"
No: "**Markdown**"


Interpretations of "probability"

[summary:  What does it *mean* to say that a fair coin has a 50% probability of landing heads? There are three common views:

- __The propensity interpretation:__ Some probabilities are just out there in the world. It's a brute fact about coins that they come up heads half the time; we'll call this the coin's physical "propensity towards heads." When we say the coin has a 50% probability of being heads, we're talking directly about this propensity.
- __The frequentist interpretation:__  When we say the coin has a 50% probability of being heads after this flip, we mean that there's a class of events similar to this coin flip, and across that class, coins come up heads about half the time. That is, the _frequency_ of the coin coming up heads is 50% inside the event class (which might be "all other times this particular coin has been tossed" or "all times that a similar coin has been tossed" etc).
- __The subjective interpretation:__ Uncertainty is in the mind, not the environment. If I flip a coin and slap it against my wrist, it's already landed either heads or tails. The fact that I don't know whether it landed heads or tails is a fact about me, not a fact about the coin. The claim "I think this coin is heads with probability 50%" is an _expression of my own ignorance,_ which means that I'd bet at 1 : 1 odds (or better) that the coin came up heads.

When the event in question is deterministic, the propensity view is commonly considered an example of the [-4yk]. The frequentist view is self-consistent, but says that one-off events (such as the probability of it raining where you are _tomorrow in particular_) cannot be assigned "probabilities", because there are no objective frequencies to draw upon. The subjectivist view extends the frequentist view, and allows you to also consider probabilities (interpreted as betting odds) on one-off events. [1bv] provides tools for reasoning about subjective probabilities in a rigorous, consistent way.]

What does it *mean* to say that a flipped coin has a 50% probability of landing heads?

Historically, there are two popular types of answers to this question, the "[frequentist_probability frequentist]" and "[4vr subjective]" (aka "[1r8 Bayesian]") answers, which give rise to  [4xx radically different approaches to experimental statistics]. There is also a third "[propensity propensity]" viewpoint which is largely discredited (assuming the coin is deterministic). Roughly, the three approaches answer the above question as follows:

- __The propensity interpretation:__ Some probabilities are just out there in the world. It's a brute fact about coins that they come up heads half the time. When we flip a coin, it has a fundamental *propensity* of 0.5 for the coin to show heads.  When we say the coin has a 50% probability of being heads, we're talking directly about this propensity.
- __The frequentist interpretation:__  When we say the coin has a 50% probability of being heads after this flip, we mean that there's a class of events similar to this coin flip, and across that class, coins come up heads about half the time. That is, the _frequency_ of the coin coming up heads is 50% inside the event class, which might be "all other times this particular coin has been tossed" or "all times that a similar coin has been tossed", and so on.
- __The subjective interpretation:__ Uncertainty is in the mind, not the environment. If I flip a coin and slap it against my wrist, it's already landed either heads or tails. The fact that I don't know whether it landed heads or tails is a fact about me, not a fact about the coin. The claim "I think this coin is heads with probability 50%" is an _expression of my own ignorance,_ and 50% probability means that I'd bet at 1 : 1 odds (or better) that the coin came up heads.

For a visualization of the differences between these three viewpoints, see [4yj]. For examples of the difference, see [4yn]. See also the [http://plato.stanford.edu/entries/probability-interpret/ Stanford Encyclopedia of Philosophy article on interpretations of probability].

The propensity view is perhaps the most intuitive view, as for many people, it just feels like the coin is intrinsically random. However, this view is difficult to reconcile with the idea that once we've flipped the coin, it has already landed heads or tails. If the event in question is decided [determinism deterministically], the propensity view can be seen as an instance of the [-4yk]: When we mentally consider the coin flip, it feels 50% likely to be heads, so we find it very easy to imagine a *world* in which the coin is _fundamentally_ 50%-heads-ish. But that feeling is actually a fact about _us,_ not a fact about the coin; and the coin has no physical 0.5-heads-propensity hidden in there somewhere — it's just a coin.

The other two interpretations are both self-consistent, and give rise to pragmatically different statistical techniques, and there has been much debate as to which is preferable. The subjective interpretation is more generally applicable, as it allows one to assign probabilities (interpreted as betting odds) to one-off events.

# Frequentism vs subjectivism

As an example of the difference between frequentism and subjectivism, consider the question: "What is the probability that Hillary Clinton will win the 2016 US presidential election?", as analyzed in the summer of 2016.

A stereotypical (straw) frequentist would say, "The 2016 presidential election only happens once.  We can't *observe* a frequency with which Clinton wins presidential elections.  So we can't do any statistics or assign any probabilities here."

A stereotypical subjectivist would say:  "Well, prediction markets tend to be pretty [well_calibrated well-calibrated] about this sort of thing, in the sense that when prediction markets assign 20% probability to an event, it happens around 1 time in 5. And the prediction markets are currently betting on Hillary at about 3 : 1 odds. Thus, I'm comfortable saying she has about a 75% chance of winning. If someone offered me 20 : 1 odds _against_ Clinton — they get \$1 if she loses, I get \$20 if she wins — then I'd take the bet. I suppose you could refuse to take that bet on the grounds that you Just Can't Talk About Probabilities of One-off Events, but then you'd be pointlessly passing up a really good bet."

A stereotypical (non-straw) frequentist would reply:  "I'd take that bet too, of course.  But my taking that bet *is not based on rigorous epistemology,* and we shouldn't allow that sort of thinking in experimental science and other important venues. You can do subjective reasoning about probabilities when making bets, but we should exclude subjective reasoning in our scientific journals, and that's what frequentist statistics is designed for. Your paper should not conclude "and therefore, having observed thus-and-such data about carbon dioxide levels, I'd personally bet at 9 : 1 odds that anthropogenic global warming is real," because you can't build scientific consensus on opinions."

...and then it starts getting complicated. The subjectivist responds "First of all, I agree you shouldn't put posterior odds into papers, and second of all, it's not like your method is truly objective — the choice of "similar events" is arbitrary, abusable, and has given rise to [https://en.wikipedia.org/wiki/Data_dredging p-hacking] and the [https://en.wikipedia.org/wiki/Replication_crisis replication crisis]." The frequentists say "well your choice of prior is even more subjective, and I'd like to see you do better in an environment where peer pressure pushes people to abuse statistics and exaggerate their results," and then [4xx down the rabbit hole we go].

The subjectivist interpretation of probability is common among artificial intelligence researchers (who often design computer systems that manipulate subjective probability distributions), wall street traders (who need to be able to make bets even in relatively unique situations), and common intuition (where people feel like they can say there's a 30% chance of rain tomorrow without worrying about the fact that tomorrow only happens once). Nevertheless, the frequentist interpretation is commonly taught in introductory statistics classes, and is the gold standard for most scientific journals.

A common frequentist stance is that it is virtuous to have a large toolbox of statistical tools at your disposal. Subjectivist tools have their place in that toolbox, but they don't deserve any particular primacy (and they aren't generally accepted when it comes time to publish in a scientific journal).

An aggressive subjectivist stance is that frequentists have invented some interesting tools, and many of them are useful, but that refusing to consider subjective probabilities is toxic. Frequentist statistics were invented in a (failed) attempt to keep subjectivity out of science in a time before humanity really understood the laws of probability theory. Now we have [1lz theorems] about how to manage subjective probabilities correctly, and how to factor personal beliefs out from the objective evidence provided by the data, and if you ignore these theorems you'll get in trouble. The frequentist interpretation is broken, and that's why science has p-hacking and a replication crisis even as all the wall-street traders and AI scientists use the Bayesian interpretation. This "let's compromise and agree that everyone's viewpoint is valid" thing is all well and good, but how much worse do things need to get before we say "oops" and start acknowledging the subjective probability interpretation across all fields of science?

The most common stance among scientists and researchers is much more agnostic, along the lines of "use whatever statistical techniques work best at the time, and use frequentist techniques when publishing in journals because that's what everyone's been doing for decades upon decades upon decades, and that's what everyone's expecting."

See also [frequentist_probability], [4vr], and [4xx].

# Which interpretation is most useful?

Probably the subjective interpretation, because it subsumes the propensity and frequentist interpretations as special cases, while being more flexible than both.

When the frequentist "similar event" class is clear, the subjectivist can take those frequencies (often called [base_rate base rates] in this context) into account. But unlike the frequentist, she can also [1lz combine those base rates with other evidence that she's seen], and assign probabilities to one-off events, and make money in prediction markets and/or stock markets (when she knows something that the market doesn't).

When the laws of physics actually do "contain uncertainty", such as when they say that there are multiple different observations you might make next with differing likelihoods (as the [schrodinger_equation Schrodinger equation] often will), a subjectivist can combine her propensity-style uncertainty with her personal uncertainty in order to generate her aggregate subjective probabilities. But unlike a propensity theorist, she's not forced to think that _all_ uncertainty is physical uncertainty: She can act like a propensity theorist with respect to Schrodinger-equation-induced uncertainty, while still believing that her uncertainty about a coin that has already been flipped and slapped against her wrist is in her head, rather than in the coin.

This fully general stance is consistent with the belief that frequentist tools are useful for answering frequentist questions: The fact that you can _personally_ assign probabilities to one-off events (and, e.g., evaluate how good a certain trade is on a prediction market or a stock market) does not mean that tools labeled "Bayesian" are always better than tools labeled "frequentist". Whatever interpretation of "probability" you use, you're encouraged to use whatever statistical tool works best for you at any given time, regardless of what "camp" the tool comes from. Don't let the fact that you think it's possible to assign probabilities to one-off events prevent you from using useful frequentist tools!


A-Class

A-Class pages are comprehensive, intuitive, engaging, well-organized, and, where it would be useful, include excellent supporting material (e.g. examples, images, and exercises). You would enjoy learning from it, and send it to your friends. To reach A-Class pages must receive detailed feedback from both the target audience to make sure it's a good experience to learn from and [5ft reviewers] with a strong understanding of the topic.

For higher level audiences asking on [4ph Slack] is a good way to get feedback, for [1r3] or [1r5] sharing with your friends (in real life of via social media) is a better bet. To get the reviewers attention ask in #reviews on Slack.

A-Class pages are:

1. Comprehensive and authoritative, covering all areas of the topic without notable omissions or inaccuracies.
2. Written to an engaging, enjoyable, professional standard.
3. Well-organized, with useful overall and local structure.
4. Aimed at an audience, with a level of technicality, jargon, and notation appropriate for them.
5. Intuitively explained, making use of examples, visual aids, requisites, and anything else which aids reader comprehension.
6. Very well-[1kl summarized], in ways appropriate for all likely audiences.

This tag should only be added by [5ft reviewers].

**[4yg Quality scale]**

* [4ym]
* [4gs]
* [72]
* [3rk]
* [4y7]
* [4yd]
* [4yf]
* [4yl]


Correspondence visualizations for different interpretations of "probability"

[summary: Let's say you have a model which says a particular coin is 70% likely to be heads. How should we assess that model?

- According to the [propensity propensity] interpretation, the coin has a fundamental intrinsic comes-up-headsness property, and the model is correct if that property is set to 0.7. (This theory is widely considered discredited, and is an example of the [-4yk]).
- According to the [frequentist_probability frequentist] interpretation, the model is saying that there are a whole bunch of different places where some similar model is saying the same thing as this one (i.e., "the coin is 70% heads"), and all models in that reference class are true if, in 70% of those different places, the coin is heads.
- According to the [4vr subjectivist] interpretation, the model is saying that the one dang coin is 70 dang percent likely to be heads, and the model is either 70% accurate (if the coin is in fact heads) or 30% accurate (if it's tails).

In other words, the propensity and frequency interpretations try to find ways to say that the model is definitively "true" or "false" (one by postulating that uncertainty is an ontologically basic part of the world, the other by identifying a collection of similar events), whereas the subjective interpretation extends the notion of "correctness" to allow for shades of gray.]

[4y9 Recall] that there are three common interpretations of what it means to say that a coin has a 50% probability of landing heads:

- __The propensity interpretation:__ Some probabilities are just out there in the world. It's a brute fact about coins that they come up heads half the time; we'll call this the coin's physical "propensity towards heads." When we say the coin has a 50% probability of being heads, we're talking directly about this propensity.
- __The frequentist interpretation:__  When we say the coin has a 50% probability of being heads after this flip, we mean that there's a class of events similar to this coin flip, and across that class, coins come up heads about half the time. That is, the _frequency_ of the coin coming up heads is 50% inside the event class (which might be "all other times this particular coin has been tossed" or "all times that a similar coin has been tossed" etc).
- __The subjective interpretation:__ Uncertainty is in the mind, not the environment. If I flip a coin and slap it against my wrist, it's already landed either heads or tails. The fact that I don't know whether it landed heads or tails is a fact about me, not a fact about the coin. The claim "I think this coin is heads with probability 50%" is an _expression of my own ignorance,_ which means that I'd bet at 1 : 1 odds (or better) that the coin came up heads.

One way to visualize the difference between these approaches is by visualizing what they say about when a model of the world should count as a good model. If a person's model of the world is definite, then it's easy enough to tell whether or not their model is good or bad: We just check what it says against the facts.  For example, if a person's model of the world says "the tree is 3m tall", then this model is [correspondence_theory_of_truth correct] if (and only if) the tree is 3 meters tall.

![ordinary truth](http://i.imgur.com/5YriFTj.jpg)

Definite claims in the model are called "true" when they correspond to reality, and "false" when they don't. If you want to navigate using a map, you had better ensure that the lines drawn on the map correspond to the territory.

But how do you draw a correspondence between a map and a territory when the map is probabilistic? If your model says that a biased coin has a 70% chance of coming up heads, what's the correspondence between your model and reality? If the coin is actually heads, was the model's claim true? 70% true? What would that mean?

![probability truth?](http://i.imgur.com/EjAto4b.jpg)

The advocate of __propensity__ theory says that it's just a brute fact about the world that the world contains ontologically basic uncertainty. A model which says the coin is 70% likely to land heads is true if and only the actual physical propensity of the coin is 0.7 in favor of heads.

![propensity correspondence](http://i.imgur.com/0vQamhR.jpg)

This interpretation is useful when the laws of physics _do_ say that there are multiple different observations you may make next (with different likelihoods), as is sometimes the case (e.g., in quantum physics). However, when the event is deterministic — e.g., when it's a coin that has been tossed and slapped down and is already either heads or tails — then this view is largely regarded as foolish, and an example of the [-4yk]: The coin is just a coin, and has no special internal structure (nor special physical status) that makes it _fundamentally_ contain a little 0.7 somewhere inside it. It's already either heads or tails, and while it may _feel_ like the coin is fundamentally uncertain, that's a feature of your brain, not a feature of the coin.

How, then, should we draw a correspondence between a probabilistic map and a deterministic territory (in which the coin is already definitely either heads or tails?)

A __frequentist__ draws a correspondence between a single probability-statement in the model, and multiple events in reality. If the map says "that coin over there is 70% likely to be heads", and the actual territory contains 10 places where 10 maps say something similar, and in 7 of those 10 cases the coin is heads, then a frequentist says that the claim is true.

![frequentist correspondence](https://i.imgur.com/RaePEL7.png)

Thus, the frequentist preserves black-and-white correspondence: The model is either right or wrong, the 70% claim is either true or false. When the map says "That coin is 30% likely to be tails," that (according to a frequentist) means "look at all the cases similar to this case where my map says the coin is 30% likely to be tails; across all those places in the territory, 3/10ths of them have a tails-coin in them." That claim is definitive, given the set of "similar cases."

By contrast, a __subjectivist__ generalizes the idea of "correctness" to allow for shades of gray. They say, "My uncertainty about the coin is a fact about _me,_ not a fact about the coin; I don't need to point to other 'similar cases' in order to express uncertainty about _this_ case. I know that the world right in front of me is either a heads-world or a tails-world, and I have a [-probability_distribution] puts 70% probability on heads." They then draw a correspondence between their probability distribution and the world in front of them, and declare that the more probability their model assigns to the correct answer, the better their model is.

![bayesian correspondence](http://i.imgur.com/OWczeTe.jpg)

If the world _is_ a heads-world, and the probabilistic map assigned 70% probability to "heads," then the subjectivist calls that map "70% accurate." If, across all cases where their map says something has 70% probability, the territory is actually that  way 7/10ths of the time, then the Bayesian calls the map "[-well_calibrated]". They then seek methods to make their maps more accurate, and better calibrated. They don't see a need to interpret probabilistic maps as making definitive claims; they're happy to interpret them as making estimations that can be graded on a sliding scale of accuracy.

## Debate

In short, the frequentist interpretation tries to find a way to say the model is definitively "true" or "false" (by identifying a collection of similar events), whereas the subjectivist interpretation extends the notion of "correctness" to allow for shades of gray.

Frequentists sometimes object to the subjectivist interpretation, saying that frequentist correspondence is the only type that has any hope of being truly objective. Under Bayesian correspondence, who can say whether the map should say 70% or 75%, given that the probabilistic claim is not objectively true or false either way? They claim that these subjective assessments of "partial accuracy" may be intuitively satisfying, but they have no place in science. Scientific reports ought to be restricted to frequentist statements, which are definitively either true or false, in order to increase the objectivity of science.

Subjectivists reply that the frequentist approach is hardly objective, as it depends entirely on the choice of "similar cases". In practice, people can (and do!) [https://en.wikipedia.org/wiki/Data_dredging abuse frequentist statistics] by choosing the class of similar cases that makes their result look as impressive as possible (a technique known as "p-hacking"). Furthermore, the manipulation of subjective probabilities is subject to the [1lz iron laws] of probability theory (which are the [ only way to avoid inconsistencies and pathologies] when managing your uncertainty about the world), so it's not like subjective probabilities are the wild west or something. Also, science has things to say about situations even when there isn't a huge class of objective frequencies we can observe, and science should let us collect and analyze evidence even then.

For more on this debate, see [4xx].


Probability interpretations: Examples

[summary: Consider evaluating, in June of 2016, the question:  "What is the probability of Hillary Clinton winning the 2016 US presidential election?"

- On the **propensity** view, Hillary has some fundamental chance of winning the election.  To ask about the probability is to ask about this objective chance.
- On the **subjective** view, saying that Hillary has an 80% chance of winning the election summarizes our *knowledge about* the election, or, equivalently, our *state of uncertainty* given what we currently know.
- On the **frequentist** view, we cannot formally or rigorously say anything about the 2016 presidential election, because it only happens once.]

## Betting on one-time events

Consider evaluating, in June of 2016, the question:  "What is the probability of Hillary Clinton winning the 2016 US presidential election?"

On the **propensity** view, Hillary has some fundamental chance of winning the election.  To ask about the probability is to ask about this objective chance.  If we see a prediction market in which prices move after each new poll — so that it says 60% one day, and 80% a week later — then clearly the prediction market isn't giving us very strong information about this objective chance, since it doesn't seem very likely that Clinton's *real* chance of winning is swinging so rapidly.

On the **frequentist** view, we cannot formally or rigorously say anything about the 2016 presidential election, because it only happens once.  We can't *observe* a frequency with which Clinton wins presidential elections.  A frequentist might concede that they would cheerfully buy for \$1 a ticket that pays \$20 if Clinton wins, considering this a favorable bet in an *informal* sense, while insisting that this sort of reasoning isn't sufficiently rigorous, and therefore isn't suitable for being included in science journals.

On the **subjective** view, saying that Hillary has an 80% chance of winning the election summarizes our *knowledge about* the election or our *state of uncertainty* given what we currently know.  It makes sense for the prediction market prices to change in response to new polls, because our current state of knowledge is changing.

## A coin with an unknown bias

Suppose we have a coin, weighted so that it lands heads somewhere between 0% and 100% of the time, but we don't know the coin's actual bias.

The coin is then flipped three times where we can see it.  It comes up heads twice, and tails once:  HHT.

The coin is then flipped again, where nobody can see it yet.  An honest and trustworthy experimenter lets you spin a wheel-of-gambling-odds,%note:The reason for spinning the wheel-of-gambling-odds is to reduce the worry that the experimenter might know more about the coin than you, and be offering you a deliberately rigged bet.% and the wheel lands on (2 : 1).  The experimenter asks if you'd enter into a gamble where you win \$2 if the unseen coin flip is tails, and pay \$1 if the unseen coin flip is heads.

On a **propensity** view, the coin has some objective probability between 0 and 1 of being heads, but we just don't know what this probability is.  Seeing HHT tells us that the coin isn't all-heads or all-tails, but we're still just guessing — we don't really know the answer, and can't say whether the bet is a fair bet.

On a **frequentist** view, the coin would (if flipped repeatedly) produce some long-run frequency $f$ of heads that is between 0 and 1.  If we kept flipping the coin long enough, the actual proportion $p$ of observed heads is guaranteed to approach $f$ arbitrarily closely, eventually.  We can't say that the *next* coin flip is guaranteed to be H or T, but we can make an objectively true statement that $p$ will approach $f$ to within epsilon if we continue to flip the coin long enough.

To decide whether or not to take the bet, a frequentist might try to apply an [unbiased_estimator unbiased estimator] to the data we have so far.  An "unbiased estimator" is a rule for taking an observation and producing an estimate $e$ of $f$, such that the [4b5 expected value] of $e$ is $f$.  In other words, a frequentist wants a rule such that, if the hidden bias of the coin was in fact to yield 75% heads, and we repeat many times the operation of flipping the coin a few times and then asking a new frequentist to estimate the coin's bias using this rule, the *average* value of the estimated bias will be 0.75.  This is a property of the _estimation rule_ which is objective.  We can't hope for a rule that will always, in any particular case, yield the true $f$ from just a few coin flips; but we can have a rule which will provably have an *average* estimate of $f$, if the experiment is repeated many times.

In this case, a simple unbiased estimator is to guess that the coin's bias $f$ is equal to the observed proportion of heads, or 2/3. In other words, if we repeat this experiment many many times, and whenever we see $p$ heads in 3 tosses we guess that the coin's bias is $rac{p}{3}$, then this rule definitely is an unbiased estimator. This estimator says that a bet of \$2 vs. $ is fair, meaning that it doesn't yield an expected profit, so we have no reason to take the bet.

On a **subjectivist** view, we start out personally unsure of where the bias $f$ lies within the interval [0, 1].  Unless we have any knowledge or suspicion leading us to think otherwise, the coin is just as likely to have a bias between 33% and 34%, as to have a bias between 66% and 67%; there's no reason to think it's more likely to be in one range or the other.

Each coin flip we see is then [22x evidence] about the value of $f,$ since a flip H happens with different probabilities depending on the different values of $f,$ and we update our beliefs about $f$ using [1zj Bayes' rule]. For example, H is twice as likely if $f=rac{2}{3}$ than if $f=rac{1}{3}$ so by [1zm Bayes's Rule] we should now think $f$ is twice as likely to lie near $rac{2}{3}$ as it is to lie near $rac{1}{3}$.

When we start with a uniform [219 prior], observe multiple flips of a coin with an unknown bias, see M heads and N tails, and then try to estimate the odds of the next flip coming up heads, the result is [21c Laplace's Rule of Succession] which estimates (M + 1) : (N + 1) for a probability of $rac{M + 1}{M + N + 2}.$

In this case, after observing HHT, we estimate odds of 2 : 3 for tails vs. heads on the next flip.  This makes a gamble that wins \$2 on tails and loses \$1 on heads a profitable gamble in expectation, so we take the bet.

Our choice of a [219 uniform prior] over $f$ was a little dubious — it's the obvious way to express total ignorance about the bias of the coin, but obviousness isn't everything. (For example, maybe we actually believe that a fair coin is more likely than a coin biased 50.<PHONE>% towards heads.) However, all the reasoning after the choice of prior was rigorous according to the laws of [1bv probability theory], which is the [probability_coherence_theorems only method of manipulating quantified uncertainty] that obeys obvious-seeming rules about how subjective uncertainty should behave.

## Probability that the 98,765th decimal digit of $\pi$ is $0$.

What is the probability that the 98,765th digit in the decimal expansion of $\pi$ is $0$?

The **propensity** and **frequentist** views regard as nonsense the notion that we could talk about the *probability* of a mathematical fact.  Either the 98,765th decimal digit of $\pi$ is $0$ or it's not.  If we're running *repeated* experiments with a random number generator, and looking at different digits of $\pi,$ then it might make sense to say that the random number generator has a 10% probability of picking numbers whose corresponding decimal digit of $\pi$ is $0$.  But if we're just picking a non-random number like 98,765, there's no sense in which we could say that the 98,765th digit of $\pi$ has a 10% propensity to be $0$, or that this digit is $0$ with 10% frequency in the long run.

The **subjectivist** considers probabilities to just refer to their own uncertainty.  So if a subjectivist has picked the number 98,765 without yet knowing the corresponding digit of $\pi,$ and hasn't made any observation that is known to them to be entangled with the 98,765th digit of $\pi,$ and they're pretty sure their friend hasn't yet looked up the 98,765th digit of $\pi$ either, and their friend offers a whimsical gamble that costs \$1 if the digit is non-zero and pays \$20 if the digit is zero, the Bayesian takes the bet.

Note that this demonstrates a difference between the subjectivist interpretation of "probability" and Bayesian probability theory. A perfect Bayesian reasoner that knows the rules of logic and the definition of $\pi$ must, by the axioms of probability theory, assign probability either 0 or 1 to the claim "the 98,765th digit of $\pi$ is a $0$" (depending on whether or not it is). This is one of the reasons why [bayes_intractable perfect Bayesian reasoning is intractable]. A subjectivist that is not a perfect Bayesian nevertheless claims that they are personally uncertain about the value of the 98,765th digit of $\pi.$ Formalizing the rules of subjective probabilities about mathematical facts (in the way that [-1bv] formalized the rules for manipulating subjective probabilities about empirical facts, such as which way a coin came up) is an open problem; this in known as the problem of [-logical_uncertainty].


Cauchy's theorem on subgroup existence: intuitive version

Cauchy's Theorem states that if $G$ is a finite [-group], and $p$ is a [4mf prime] dividing the [3gg order] of $G$, then $G$ has an element of order $p$.
Equivalently, it has a subgroup of order $p$.

This theorem is very important as a partial converse to [4jn Lagrange's theorem]: while Lagrange's theorem gives us *restrictions* on what subgroups can exist, Cauchy's theorem tells us that certain subgroups *definitely do* exist.
In this direction, Cauchy's theorem is generalised by the [sylow_theorems_on_subgroup_existence Sylow theorems].

# Proof

Let $G$ be a group, and write $e$ for its identity.
We will try and find an element of order $p$.

What does it mean for an element $x
ot = e$ to have order $p$? Nothing more nor less than that $x^p = e$.
(This is true because $p$ is prime; if $p$ were not prime, we would additionally have to stipulate that $x^i$ is not $e$ for any smaller $i < p$.)

Now follows the magic step which casts the problem in the "correct" light.

Consider all possible combinations of $p$ elements from the group.
For example, if $p=5$ and the group elements are $\{ a, b, c, d, e\}$ with $e$ the identity, then $(e, e, a, b, a)$ and $(e,a,b,a,e)$ are two different such combinations.
Then $x
ot = e$ has order $p$ if and only if the combination $(x, x, \dots, x)$ multiplies out to give $e$.

The rest of the proof will be simply following what this magic step has told us to do.

Since we want to find some $x$ where $(x, x, \dots, x)$ multiplies to give $e$, it makes sense only to consider the combinations which multiply out to give $e$ in the first place.
So, for instance, we will exclude the tuple $(e, e, a, b, a)$ from consideration if it is the case that $eeaba$ is the identity (equivalently, that $aba = e$).
For convenience, we will label the set of all the allowed combinations: let us call it $X$.

Of the tuples in $X$, we are looking for one with a very specific property: every entry is equal.
For example, the tuple $(a,b,c,b,b)$ is no use to us, because it doesn't tell us an element $x$ such that $x^p = e$; it only tells us that $abcbb = e$.
So we will be done if we can find a tuple in $X$ which has every element equal (and which is not the identity).

We don't really have anything to go on here, but it will surely help to know the size of $X$: it is $|G|^{p-1}$, since every element of $X$ is determined exactly by its first $p-1$ places (after which the last is fixed).
There are no restrictions on the first $p-1$ places, though: we can find a $p$th element to complete any collection of $p-1$.

%%hidden(Example):
If $p=5$, and we have the first four elements $(a, a, b, e, \cdot)$, then we know the fifth element *must* be $b^{-1} a^{-2}$.
Indeed, $aabe(a^{-1} a^{-2}) = e$, and [ inverse elements of a group are unique], so nothing else can fill the last slot.
%%

So we have $X$, of size $|G|^{p-1}$; we have that $p$ divides $|G|$ (and so it divides $|X|$); and we also have an element $(e,e,\dots,e)$ of $X$.
But notice that if $(a_1, a_2, \dots, a_p)$ is in $X$, then so is $(a_2, a_3, \dots, a_p, a_1)$; and so on through all the rotations of an element.
We can actually group up the elements of $X$ into buckets, where two tuples are in the same bucket if (and only if) one can be rotated to make the other.

How big is a bucket? If a bucket contains something of the form $(a, a, \dots, a)$, then of course that tuple can only be rotated to give one thing, so the bucket is of size $1$.
But if the bucket contains any tuple $T$ which is not "the same element repeated $p$ times", then there must be exactly $p$ things in the bucket: namely the $p$ things we get by rotating $T$. (Exercise: verify this!)

%%hidden(Show solution):
The bucket certainly does contain every rotation of $T$: we defined the bucket such that if a tuple is in the bucket, then so are all its rotations.
Moreover, everything in the bucket is a rotation of $T$, because two tuples are in the bucket if and only if we can rotate one to get the other; equivalently, $A$ is in the bucket if and only if we can rotate it to get $T$.

How many such rotations are there? We claimed that there were $p$ of them.
Indeed, the rotations are precisely $$(a_1, a_2, \dots, a_p), (a_2, a_3, \dots, a_p, a_1), \dots, (a_{p-1}, a_p, a_1, \dots, a_{p-2}), (a_p, a_1, a_2, \dots, a_{p-1})$$
and there are $p$ of those; are they all distinct?
Yes, they are, and this is because $p$ is prime (it fails when $p=8$, for instance, because $(1,1,2,2,1,1,2,2)$ can be rotated four places to give itself).
Indeed, if we could rotate the tuple $T$ nontrivially (that is, strictly between $1$ and $p$ times) to get itself, then the integer $n$, the "number of places we rotated", must divide the integer "length of the tuple". [todo: explain this better]
But that tells us that $n$ divides the prime $p$, so $n$ is $1$ or $p$ itself, and so either the tuple is actually "the same element repeated $p$ times" (in the case $n=1$) %%note: But we're only considering $T$ which is not of this form!%%, or the rotation was the "stupid" rotation by $p$ places (in the case $n=p$).
%%

OK, so our buckets are of size $p$ exactly, or size $1$ exactly.
We're dividing up the members of $X$ - that is, $|G|^{p-1}$ things - into buckets of these size; and we already have one bucket of size $1$, namely $(e,e,\dots,e)$.
But if there were no more buckets of size $1$, then we would have $p$ dividing $|G|^{p-1}$ and also dividing the size of all but one of the buckets.
Mixing together all the other buckets to obtain an uber-bucket of size $|G|^{p-1} - 1$, the total must be divisible by $p$, since each individual bucket was %%note:Compare with the fact that the sum of even numbers is always even; that is the case that $p=2$.%%; so $|G|^{p-1} - 1$ is divisible by $p$.
But that is a contradiction, since $|G|^{p-1}$ is also divisible by $p$.

So there is another bucket of size $1$.
A bucket of size $1$ contains something of the form $(a,a,\dots,a)$: that is, it has shown us an element of order $p$.