Conjugacy classes, characters and products of elements

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.

Non-abelian anyons and some cousins of the Arad–Herzog conjecture

“…One natural question that remains is to better understand to what extent ideas involving non-abelian anyons can be used to prove the AH conjecture (see [24][25][26] and references therein for interesting recent progress on the AH conjecture). Since discrete gauge theories feature in various physical systems, perhaps we can hope for a physics proof of this conjecture.…”

Criteria for solubility and nilpotency of finite groups with automorphisms

“…Obviously the condition above is also necessary for the nilpotency of $$G$$. We mention that there are several recent results related to the theorem of Baumslag and Wiegold (see, for example, [2, 3, 6, 10, 12]).…”

Criteria for solubility and nilpotency of finite groups with automorphisms

“…Here the symbol |x| stands for the order of an element x in a group G. Obviously the condition above is also necessary for the nilpotency of G. We mention that there are several recent results related to the theorem of Baumslag and Wiegold (see for example [2,3,10,12,6]).…”