MA3343 Groups. Week 6: Conjugacy and Conjugacy Classes
Welcome to our activities for Week 6 (the week of November 6).
We have another major concept this week, closely related to the centralizer of an element which we discussed last week. This week's theme is conjugacy.
In our first video lecture, we will define what it means for two elements in a group to be conjugate to each other, we'll see that conjugacy is an equivalence relation that splits the group into disjoint subsets called conjugacy classes.
In the second one, we'll figure out an important relation in finite groups between the number of distinct conjugates of a given element and the order of its centralizer. This will make use of the concept of left cosets, which we encountered in the proof of Lagrange's Theorem.
We are still in
Section 2.2 of the lecture notes.
Next week we'll move on to discuss the meaning of conjugacy in the particular case of symmetric groups (groups of permutations). The context there is a bit more concrete than the abstract reasoning of the last couple of weeks.