# 3-cycle in symmetric group:S3

We consider the group $G$ defined as symmetric group:S3, i.e., the symmetric group of degree three, which for convenience we take to be the symmetric group acting on the set $\{ 1, 2, 3 \}$.

We are interested in the conjugacy class of 3-cycles in this group, i.e., permutations that move all elements and have order three. The 3-cycles form a single conjugacy class and also form a single orbit under the action of the automorphism group of $S_3$.

The full list of elements in the conjugacy class is: $\{ (1,2,3), (1,3,2) \}$

## Arithmetic functions

Function Value Explanation
order of the whole group 6
size of conjugacy class 2
size of automorphism class 2
number of conjugacy classes in automorphism class 1
order of elements in conjugacy class 3

## Description in alternative interpretations of the whole group

Interpretation of $G$ Description of conjugacy class Verification of size computation Verification of element order computation More information
As the symmetric group of degree three corresponds to the partition 3 $\frac{3!}{3} = 2$ $\operatorname{lcm}\{ 3 \} = 3$ element structure of symmetric groups, element structure of symmetric group:S3#Interpretation as symmetric group, conjugacy class size formula in symmetric group
As the dihedral group $D_{2n}$, $n = 3, 2n = 6$ ( $n$ odd) the generator of the cyclic piece of order $n$ 2 (valid for $n \ge 3$) $n = 3$ element structure of dihedral groups, element structure of symmetric group:S3#Interpretation as dihedral group
As the special linear group $SL(2,q)$, $q = 2$ ( $q$ even) the unique conjugacy class of semisimple elements not diagonalizable over the field itself $q(q - 1) = 2(2 - 1) = 2$ Must divide $q^2 - 1$ in general, in this case, it is $q^2 - 1 = 2^2 - 1 = 3$ element structure of special linear group of degree two over a finite field, element structure of symmetric group:S3#Interpretation as general linear group of degree two
As the general affine group $GA(1,q)$, $q = 3$ the conjugacy class of all maps of the form $x \mapsto x + b, b \in \mathbb{F}_3^\ast$. $q - 1 = 2$ $q = 3$ element structure of general affine group of degree one over a finite field, element structure of symmetric group:S3#Interpretation as general affine group of degree one