1,908 research outputs found
Simple groups admit Beauville structures
We answer a conjecture of Bauer, Catanese and Grunewald showing that all
finite simple groups other than the alternating group of degree 5 admit unmixed
Beauville structures. We also consider an analog of the result for simple
algebraic groups which depends on some upper bounds for character values of
regular semisimple elements in finite groups of Lie type and obtain definitive
results about the variety of triples in semisimple regular classes with product
1. Finally, we prove that any finite simple group contains two conjugacy
classes C,D such that any pair of elements in C x D generates the group.Comment: 30 pages, in the second version, some results are improved and in
particular we prove an irreducibility for a certain variet
Products of conjugacy classes and fixed point spaces
We prove several results on products of conjugacy classes in finite simple
groups. The first result is that there always exists a uniform generating
triple. This result and other ideas are used to solve a 1966 conjecture of
Peter Neumann about the existence of elements in an irreducible linear group
with small fixed space. We also show that there always exist two conjugacy
classes in a finite non-abelian simple group whose product contains every
nontrivial element of the group. We use this to show that every element in a
non-abelian finite simple group can be written as a product of two rth powers
for any prime power r (in particular, a product of two squares).Comment: 44 page
Some Exceptional Beauville Structures
We first show that every quasisimple sporadic group possesses an unmixed
strongly real Beauville structure aside from the Mathieu groups M11 and M23
(and possibly 2B and M). We go on to show that no almost simple sporadic group
possesses a mixed Beauville structure. We then go on to use the exceptional
nature of the alternating group A6 to give a strongly real Beauville structure
for this group explicitly correcting an earlier error of Fuertes and
Gonzalez-Diez. In doing so we complete the classification of alternating groups
that possess strongly real Beauville structures. We conclude by discussing
mixed Beauville structures of the groups A6:2 and A6:2^2.Comment: v4 - case Co2 ammende
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