# What can you say about a finite group if you know something about the conjugacy class sizes?

The influence of the sizes of conjugacy classes of a finite group.

Let G be a finite group.

Question 1 Given information about the sizes of the conjugacy classes of the elements of a group what can we say about the group?

As stated the question is a little vague but let me give some very early answers:

Answer 1 (Sylow (1870)) If all the sizes are a power of a prime p then G has a non-trivial centre.

Answer 2 (Burnside (1904)) If at least one size is the power of a prime then G is not simple.

So we have some quite old answers; although the first answer is straightforward to prove the second is quite hard.

There has been a lot of activity on these problems in recent years and in this lecture I would like to touch on what, I think, are some of the more interesting.

Here is one result I will mention.

Theorem 1 (Camina and Camina) Let G be a finite group with the property that given any three distinct conjugacy class sizes greater than 1 there is a pair which is coprime. Then G has at most three conjugacy class sizes greater than 1, and G is soluble.

In this talk both the background to the result and some interesting open questions that arise will be discussed. This will be based on the survey recently written jointly with Rachel Camina.

This talk is part of the Algebra seminar series.