Products of Conjugacy Classes

Wednesday, October 1, 2014 - 11:30am - 12:20pm
Keller 3-180
Bob Guralnick (University of Southern California)
Let G be a finite or algebraic group. Given a multiset of conjugacy classes of G, we are interested in the product of such. This is some normal subset of G. In particular, one is interested in the size of this product and when is the product all of G. There are some nice results about the asymptotics of this situation, but we will focus on the case of two conjugacy classes. In particular if C, D are conjugacy classes of G, what is CD and when is CD=G. The key to studying this problem is a basic formula using character theory (and is not unrelated to Gower's triple product theorem). We will discuss some interesting consequences using this setup including finite simple groups admitting Beauville surfaces, fixed point spaces of elements in linear groups, products of centralizers in simple finite and algebraic groups, the Arad-Herzog conjecture and a generalization of the Baer-Suzuki theorem. On the other hand, the state of knowledge here is far from complete.
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