On the Diameters of Commuting Graphs of Permutational Wreath Products

Abstract

Let G be a group and let Z(G) be the center of G. The commuting graph of the group G is an undirected graph Γ(G) with the vertex set G \ Z(G) such that two vertices x, y are adjacent if and only if xy = yx. We study the commuting graphs of permutational wreath products H G, where G is a transitive permutation group acting on X (the top group of the wreath product) and (H, Y) is an Abelian permutation group acting on Y.