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Fixed point ratios for primitive groups and applications

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Let G be a finite permutation group. The fixed point ratio of an element x in G, denoted fpr(x), is the proportion of points fixed by x. Fixed point ratios for finite primitive groups have been studied for many decades, finding a wide range of applications. In this talk, I will discuss some recent joint work with Bob Guralnick where we determine the triples (G,x,r) such that G is primitive, x has prime order r and fpr(x)>1/(r+1). This turns out to have some interesting applications and we can use it to obtain new results on the minimal degree and minimal index of primitive groups. Another application arises in joint work with Moreto and Navarro on the commuting probability of p-elements in finite groups.

This talk is part of the Algebra seminar series.

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