Groups Week 7 22/23

Conjugacy in Symmetric Groups

Welcome to our activities for Week 7 (the week of October 17).

Our topic this week is the meaning of conjugacy in the special situation of symmetric groups. The relevant section of the lecture notes is Section 3.1. 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 Sn is the group of all permutations of n objects, under the operation of function composition.

Plans for this week:
  1. In our first lecture, we will describe the representation of a permutation as a product of disjoint cycles. You may already know about this, but maybe not about the connection to conjugacy in symmetric groups.
  2. In our second lecture, we will think about the numbers of elements in each conjugacy class, and the orders of centralizers of elements of $S_n$.
  3. The second homework sheet is now open on the Blackboard page. The last few problems there (not the very last one) are related to this week’s content.
  4. Slides for this week’s lectures
  5. Video versions are here, in case they are of use.