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The spread of a finite group

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Let G be a finite group. We say that G has spread at least k if for any k distinct nontrivial elements x1,...,xkG, there exists yG such that xi and y generate G for every i=1,...,k. If G does not have spread at least 1 then G is said to have spread 0. Using elementary methods we can prove that if G has a non-trivial normal subgroup N such that G/N is non-cyclic then G must have spread 0. It has been conjectured by Guralnick and Kantor that the converse is true. They can prove that the converse holds in many cases. We will discuss some recent joint work with Tim Burness involving the remaining cases.

This talk is part of the Algebra Seminar series.

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