Conjugacy classes of the symmetric group on five elements

https://arbital.com/p/symmetric_group_five_conjugacy_classes

by Patrick Stevens Jun 14 2016 updated Jun 15 2016

The symmetric group on five elements is a group of just the right size to make a good example of a table of conjugacy classes.


The Symmetric group on five generators has size (where the exclamation mark denotes the Factorial function). By the result that in a symmetric group, conjugacy classes coincide with cycle types, we can list the conjugacy classes of easily.

The table

Determining the list of cycle types and sizes

There are five elements to permute; we need to find all the possible ways of grouping them up into disjoint cycles. We will go through this systematically. Note first that since there are only five elements to permute, there cannot be any element with a -cycle or higher.

Those with largest cycle of length

If there is a -cycle, then it permutes everything, so its cycle type is . That is, we can take a representative , and this is the only conjugacy class with a -cycle.

Recall that every cycle of length may be written in five different ways: or or , and so on. Without loss of generality, we may assume comes at the start (by cycling round the permutation if necessary).

Then there are ways to fill in the remaining slots of the cycle (where the exclamation mark denotes the Factorial function).

Hence there are elements of this conjugacy class.

Those with largest cycle of length

If there is a -cycle, then it permutes everything except one element, so its cycle type must be (abbreviated as ). That is, we can take a representative , and this is the only conjugacy class with a -cycle.

Either the element is permuted by the -cycle, or it is not.

This comes to a total of possible -cycles in this conjugacy class.

Those with largest cycle of length

Now we have two possible conjugacy classes: and .

The class

An example representative for this class is .

We proceed with a slightly different approach to the case. Using the notation for the [-binomial_coefficient], we have possible ways to select the numbers which appear in the -cycle. Each selection has two distinct ways it could appear as a -cycle: the selection can appear as (or the duplicate cycles and ), or as (or the duplicate cycles or ).

That is, we have elements of this conjugacy class.

The class

An example representative for this class is .

Again, there are possible ways to select the numbers which appear in the -cycle; having made this selection, we have no further choice about the -cycle.

Given a selection of the elements of the -cycle, as before we have two possible ways to turn it into a -cycle.

We are also given the selection of the elements of the -cycle, but there is no choice about how to turn this into a -cycle because, for instance, is equal to as cycles. So this time the selection of the elements of the -cycle has automatically determined the corresponding -cycle.

Hence there are again elements of this conjugacy class.

Those with largest cycle of length

There are two possible cycle types: and .

The class

An example representative for this class is .

There are ways to select the first two elements; then once we have done so, there are ways to select the second two. Having selected the elements moved by the first -cycle, there is only one distinct way to make them into a -cycle, since (for example) is equal to as permutations; similarly the selection of the elements determines the second -cycle.

However, this time we have double-counted each element, since (for example) the permutation is equal to by the result that disjoint cycles commute.

Therefore there are elements of this conjugacy class.

The class

An example representative for this class is .

There are ways to select the two elements which this cycle permutes. Once we have selected the elements, there is only one distinct way to put them into a -cycle, since (for example) is equal to as permutations.

Therefore there are elements of this conjugacy class.

Those with largest cycle of length

There is only the identity in this class, so it is of size .