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CATEGORIES:Birmingham and Warwick Algebra Seminar
SUMMARY:What can you say about a finite group if you know
something about the conjugacy class sizes? - Alan
Camina\, University of East Anglia
DTSTART:20121101T160000Z
DTEND:20121101T170000Z
UID:TALK2563AT
URL:/talk/index/2563
DESCRIPTION:*The influence of the sizes of conjugacy classes o
f a finite group.*\n\nLet _G_ be a finite group.\n
\n*Question 1* Given information about the sizes o
f the conjugacy classes of the elements of a group
what can we say about the group?\n\nAs stated the
question is a little vague but let me give some v
ery early answers:\n\n*Answer 1* (Sylow (1870)) If
all the sizes are a power of a prime _p_ then _G_
has a non-trivial centre.\n\n*Answer 2* (Burnside
(1904)) If at least one size is the power of a pr
ime then _G_ is not simple.\n\nSo we have some qui
te old answers\; although the first answer is stra
ightforward to prove the second is quite hard.\n\n
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.\n\nHere is one result I will mention.
\n\n*Theorem 1* (Camina and Camina) Let _G_ be a f
inite group with the property that given any three
distinct conjugacy class sizes greater than 1 the
re is a pair which is coprime. Then _G_ has at mos
t three conjugacy class sizes greater than 1\, and
_G_ is soluble.\n\nIn this talk both the backgrou
nd to the result and some interesting open questio
ns that arise will be discussed. This will be base
d on the survey recently written jointly with Rach
el Camina.
LOCATION:Watson Building\, Lecture Room C
CONTACT:David Craven
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