Element structure of symmetric groups

This article gives specific information, namely, element structure, about a family of groups, namely: symmetric group.
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The symmetric group on a set is the group, under multiplication, of permutations of that set. The symmetric group of degree $n$ is the symmetric group on a set of size $n$. For convenience, we consider the set to be $\{1,2,\dots ,n\}$.

This article discusses the element structre of the symmetric group of degree $n$.

Conjugacy class structure and cycle type

General result

Further information: Cycle type, cycle type determines conjugacy class

The cycle type of a permutation on a set of size $n$ is defined as the corresponding unordered integer partition of $n$ into the sizes of the cycles in the cycle decomposition. For instance, the permutation $(1,2,3,4,5)(6,7,8)(9,10,11,12)$ has cycle type $5+3+4$.

It turns out that there is a bijection between the set of conjugacy classes in the symmetric group of degree $n$ and the set of unordered integer partitions via the cycle type map, because cycle type determines conjugacy class.

The size of a conjugacy class corresponding to a cycle type with $a_j$ parts of size $j$, is:

$\frac{n!}{{\prod }_j{a}_j!(j{)}^{a_j}}$

Particular cases

Degree	Number of conjugacy classes	List of conjugacy class sizes	Pairs of (partition,conjugacy class)
$1$	$1$	$1$	$(1,1)$
$2$	$2$	$1,1$	$(2,1),(1+1,1)$
$3$	$4$	$1,2,3$	$(1+1+1,1),(3,2),(2+1,3)$
$4$	$5$	$1,3,6,6,8$	$(1+1+1+1,1),(2+2,3),(4,6),(2+1+1,6),(3+1,8)$
$5$	$7$	$1,10,15,20,20,24,30$	$(1+1+1+1+1,1),(2+1+1+1,10),(2+2+1,15),(3+2,20),(3+1+1,20),(5,24),(4+1,30)$