Products of conjugacy classes and fixed point spaces

2010/05/20 by Guralnick, Robert, Malle, Gunter · 2 citations
#20C15 (Secondary) #20C20 #20D05 (Primary) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1005.3756

Abstract

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).

Cited by

• Minimal generation of finite simple groups of Lie type by regular unipotent elements
• Fixed Point Free Actions of Group Elements in Linear Groups and Finite Group Representations

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