Groups in which the centralizer of any non-central primary element is maximal

Abstract

Abstract We investigate the structure of finite group G in which the centralizer of each non-central primary element of G is maximal in G. This provides an answer to the question raised in [X. Zhao, R. Chen and X. Guo, Groups in which the centralizer of any non-central element is maximal, J. Group Theory 23 2020, 5, 871–878], and also is a generalization of the result in [A. Mann, Finite groups with maximal normalizers, Illinois J. Math. 12 1968, 67–75]. In particular, we give an independent criterion of simplicity of a group, asserting that a group G is simple if the centralizer of each non-central primary element is maximal in G.