Element structure of dihedral groups

This article gives specific information, namely, element structure, about a family of groups, namely: dihedral group.
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This article discusses the element structure of the dihedral group $D_{2n}$ of degree $n$ and order $2n$, given by the presentation:

$⟨x,a\mid a^n=x^2=e,xa{x}^{-1}=a^{-1}⟩$

Here $e$ denotes the identity element.

Summary

The information below is for $D_{2n}$.

Item	Value
order	$2n$
conjugacy class sizes	Case $n$ odd: 1 (1 time), 2 ($(n-1)/2$ times), $n$ (1 time) Case $n$ even: 1 (2 times), 2 ($(n-2)/2$ times), $n/2$ (2 times)
number of conjugacy classes	$(n+3)/2$ if $n$ odd, $(n+6)/2$ if $n$ even
number of orbits under automorphism group	$d(n)+1$ where $d$ is the divisor count function if $n\ge 3$, 4 if $n=2$
order statistics	Case $n$ odd: $\phi (d)$ of order $d$ for $d|n$, $n$ of order 2

Odd degree case

This is the case $n$ is odd, so $2n$ is twice an odd number.

Conjugacy class structure

Nature of conjugacy class	Size of each conjugacy class	Number of such conjugacy classes	Total number of elements
Identity element	1	1	1
Non-identity elements in cyclic subgroup $⟨a⟩$, where each element and its inverse form a conjugacy class	2	$(n-1)/2$	$n-1$
Elements outside the cyclic subgroup $⟨a⟩$, all form a single conjugacy class	$n$	1	$n$
Total	--	$(n+3)/2$ (number of conjugacy classes)	$2n$ (order of group)

Order information

We have the following number of elements of various orders:

Order type	Number of elements of that order	Specifics about the elements	Number of such order types	Total number of elements
$d$ a divisor of $n$ (hence $d$ is odd)</math>	$\phi (d)$, $\phi$ is the Euler totient function	These are all the generators of the cyclic subgroup $⟨a^{n/d}⟩$	$d(n)$ where $d$ is the divisor count function.	${\sum }_{d|n}\phi (d)$ which becomes $n$
2	$n$	All the elements outside $⟨a⟩$	1	$n$

Equivalence classes up to automorphism

It turns out that the equivalence classes up to automorphism are given precisely by order of elements, i.e., two elements are automorphic elements if and only if they have the same order. Equivalently, two elements are automorphic if and only if they generate the same cyclic subgroup. More explicitly:

Description of equivalence class	Number of conjugacy classes per equivalence class	Size of each conjugacy class	Total number of elements in each equivalence class	Number of equivalence classes	Total number of conjugacy classes	Total number of elements across equivalence classes
Identity element	1	1	1	1	1	1
For each nontrivial divisor $d$ of $n$, the elements of order $d$ (generators of $⟨a^{n/d}$) form one equivalence class	$\phi (d)/2$	2	$\phi (d)$, $\phi$ is the Euler totient function	$d(n)-1$, $d(n)$ is the divisor count function	$(n-1)/2$	${\sum }_{d|n,d>1}\phi (d)=n-1$
Elements outside the cyclic subgroup $⟨a⟩$, of order 2	1	$n$	$n$	1	1	$n$
Total	--	--	--	$d(n)+1$	$(n+3)/2$	$2n$

Even degree case

Suppose $n=2m$, and $D_{2n}$ is the dihedral group of order $2n$. Then, $D_{2n}$ has the following conjugacy classes (a total of $(n+6)/2$ conjugacy classes):

Conjugacy class structure

Nature of conjugacy class	Size of each conjugacy class	Number of such conjugacy classes	Total number of elements
Identity element	1	1	1
Non-identity element $a^{n/2}$ of order 2 in $⟨a⟩$	1	1	1
Non-identity elements in cyclic subgroup $⟨a⟩$, where each element and its inverse form a conjugacy class	2	$(n-2)/2$	$n-2$
Elements outside $⟨a⟩$, form two conjugacy classes, one for elements of the form $a^{2k}x$, and the other for elements of the form $a^{2k+1}x$	$n/2$	2	$n$
Total	--	$(n+6)/2$ (number of conjugacy classes)	$2n$ (order of group)

Order information

We have the following number of elements of various orders:

Order	Number of elements of that order	Specifics about the elements
$d$ a divisor of $n$ other than 2	$\phi (d)$, $\phi$ is the Euler totient function	These are all the generators of the cyclic subgroup $⟨a^{n/d}⟩$
2	$n+1$	All the elements outside $⟨a⟩$, as well as the element $a^{n/2}$

Equivalence classes up to automorphism

Note that the data in the table below is not correct for the case $n=2$, in which case we get the Klein four-group.

The equivalence classes up to automorphism are as follows:

Description of equivalence class	Number of conjugacy classes per equivalence class	Size of each conjugacy class	Total number of elements in each equivalence class	Number of equivalence classes	Total number of conjugacy classes	Total number of elements across equivalence classes
Identity element	1	1	1	1	1	1
Element $a^{n/2}$	1	1	1	1	1	1
For each nontrivial divisor $d$ of $n$ other than 2, the elements of order $d$ (generators of $⟨a^{n/d}$) form one equivalence class	$\phi (d)/2$	2	$\phi (d)$, $\phi$ is the Euler totient function	$d(n)-2$, $d(n)$ is the divisor count function	$(n-2)/2$	${\sum }_{d|n,d>2}\phi (d)=n-2$
Elements outside the cyclic subgroup $⟨a⟩$, of order 2	2	$n/2$	$n$	1	2	$n$
Total	--	--	--	$d(n)+1$	$(n+6)/2$	$2n$