MA3343 Groups. Week 6: Conjugacy in Symmetric Groups

Welcome to our activities for Week 7 (the week of November 9).

Our topic this week is the meaning of conjugacy in the special situation of symmetric groups. Last week, we defined the abstract concept of conjugacy in any group, and noted that every group is the disjoint union of its conjugacy classes, and that (in a finite group) the number of elements in the conjugacy class of a particular element is the index of its centralizer. We also noted that for matrices in a general linear group, conjugacy is the same as the relation of similarity in linear algebra; two matrices are conjugate to each other if and only if they represent the same linear transformation, possibly with respect to different bases. In general, given a pair of elements in a group, there is not really a fast way to figure out if they are conjugates of each other. An exception to that is the case of the symmetric groups, where there is a nice combinatorial description of conjugacy classes. Recall from Section 1.1 that the symmetric group of degree n, denoted S_n is the group of all permutations of n objects, under the operation of function composition.

In the first of this week's video lectures, we describe the representation of a permutation as a product of disjoint cycles. This one is a bit longer than usual, sorry for that. You may already be familiar with some of this content.

Accompanying slides are here (annotated version) and here (without annotation).

In the second video, we observe that two elements of S_n are conjugate to each other in S_n if and only if they have the same cycle type. This fact is of great interest to group theorists and combinatorialists.

Accompanying slides are here (annotated) andhere (without annotation).

Lecture Notes

This week's content is in Section 2.3 of the lecture notes, which as usual contain some extra detail not included in the video explanations.