Finite groups whose noncentral commuting elements have centralizers of equal size

Abstract

A finite group is called a CH-group if for every x,yGZ(G), xy=yx implies that ∥CG(x)∥ ∥CG(y)∥. Applying results of Schmidt [Zentralisatorverbnde endlicher Gruppen, Rend. Sem. Mat. Univ. Padova 44 (1970), 97-131] and Rebmann [F-Gruppen, Arch. Math.22 (1971), 225-230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CH-groups. © 2010 Australian Mathematical Publishing Association Inc.