Groups with anticentral elements

Abstract

We study finite groups $G$ with elements $g$ such that $|{\mathbf{C}}_G(g)|=|G:G^{\prime }|$. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class of nilpotent supplements for the commutator subgroup and, using the classification of finite simple groups, that these groups are solvable.

12 pages; old paper from 2008 put on arXiv for the sake of "green" open access