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CATEGORIES:Junior Algebra and Number Theory seminar
SUMMARY:The Influence of Conjugacy Class sizes on Sylow Su
bgroups - Julian Brough University of Cambridge
DTSTART;TZID=Europe/London:20131023T150000
DTEND;TZID=Europe/London:20131023T160000
UID:TALK48561AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/48561
DESCRIPTION:Given a group G and x in G\, the size of the conju
gacy class of x in G is given by the size of the g
roup divided by the order. This number will be ref
ered to as the index of x.\nIn the subject of repr
esentation theory\, conjugacy class sizes form a k
ey component in the construction\nof the character
table of a group\, for example in the orthogonali
ty relations. The character table then enables us
to determine group structures such as normal subgr
oups or see how conjugacy classes multiply togethe
r. Hence it is natural to ask what information can
be obtained about a group from the class sizes. A
s an example one of Burnside's theorems states a
finite group with an index which is a prime power
can not be simple.\nGiven a group G\, let cl(G) de
note the set of conjugacy class sizes of G. As a c
ase of Burnside's theorem\,\nA. Camina considered
cl(G) which is the product of two prime powers\, a
nd 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 any p element x of G\, C_G(x
) contains a Sylow p subgroup. Which is the same a
s saying x has p' index. However is the converse\n
to this statement true\, i.e. let G be a group\, i
f all p elements of G have p' index\, does P have
to be abelian?
LOCATION:CMS\, MR5
CONTACT:Julian Brough
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