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SUMMARY:The Influence of Conjugacy Class sizes on Sylow Subgroups - Julian
  Brough University of Cambridge
DTSTART:20131023T140000Z
DTEND:20131023T150000Z
UID:TALK48561@talks.cam.ac.uk
CONTACT:Julian Brough
DESCRIPTION:Given a group G and x in G\, the size of the conjugacy class o
 f x in G is given by the size of the group divided by the order. This numb
 er will be refered to as the index of x.\nIn the subject of representation
  theory\, conjugacy class sizes form a key component in the construction\n
 of the character table of a group\, for example in the orthogonality relat
 ions. The character table then enables us to determine group structures su
 ch as normal subgroups or see how conjugacy classes multiply together. Hen
 ce it is natural to ask what information can be obtained about a group fro
 m the class sizes. As an example one of Burnside's theorems states a fini
 te group with an index which is a prime power can not be simple.\nGiven a 
 group G\, let cl(G) denote the set of conjugacy class sizes of G. As a cas
 e of Burnside's theorem\,\nA. Camina considered cl(G) which is the product
  of two prime powers\, and showed the group is nilpotent.\nLet G be a grou
 p\, p a prime and P a Sylow p subgroup of G. If P is abelian\, then for an
 y 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\nto this statement true
 \, i.e. let G be a group\, if all p elements of G have p' index\, does P h
 ave to be abelian?
LOCATION:CMS\, MR5
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