FINITE GROUPS WHOSE NONCENTRAL COMMUTING ELEMENTS HAVE CENTRALIZERS OF EQUAL SIZE

Abstract

Abstract A finite group is called a CH -group if for every x , y ∈ G ∖ Z ( G ), xy = yx implies that $\|\cent Gx\| = \|\cent Gy\|$ . Applying results of Schmidt [‘Zentralisatorverbände 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.