The Influence of Conjugacy Class sizes on Sylow Subgroups

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• Julian Brough University of Cambridge
• Wednesday 23 October 2013, 15:00-16:00
• CMS, MR5.

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Given a group G and x in G, the size of the conjugacy class of x in G is given by the size of the group divided by the order. This number will be refered to as the index of x. In the subject of representation theory, conjugacy class sizes form a key component in the construction of the character table of a group, for example in the orthogonality relations. The character table then enables us to determine group structures such as normal subgroups or see how conjugacy classes multiply together. Hence it is natural to ask what information can be obtained about a group from the class sizes. As an example one of Burnside’s theorems states a finite group with an index which is a prime power can not be simple. Given a group G, let cl(G) denote the set of conjugacy class sizes of G. As a case of Burnside’s theorem, A. Camina considered cl(G) which is the product of two prime powers, and showed the group is nilpotent. Let G be a group, p a prime and P a Sylow p subgroup of G. If P is abelian, then for any p element x of G, C_G(x) contains a Sylow p subgroup. Which is the same as saying x has p’ index. However is the converse to this statement true, i.e. let G be a group, if all p elements of G have p’ index, does P have to be abelian?

This talk is part of the Junior Algebra and Number Theory seminar series.

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