Conjugacy classes, characters and products of elements

Robert M. Guralnick and Alexander Moretó

Mathematische Nachrichten, 2019, vol. 292, issue 6, 1315-1320

Abstract: Recently, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy)=o(x)o(y) for every x,y∈G of coprime order. Motivated by this result, we study the groups with the property that (xy)G=xGyG and those with the property that χ(xy)=χ(x)χ(y) for every χ∈Irr(G) and every nontrivial x,y∈G of pairwise coprime order. We also consider several ways of weakening the hypothesis on x and y. While the result of Baumslag and Wiegold is completely elementary, some of our arguments here depend on (parts of) the classification of finite simple groups.

Date: 2019
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