A Characteristic Property of Finite p -Nilpotent Groups

Abstract

An element x of a group G is called Q-central if there exists a central chief factor H/K of G such that x ∈ H\K. It is proved that a finite group G is p-nilpotent if and only if every element in G _p  \Φ(G _p ) is Q-central. We will adapt Doerk and Hawkes Doerk, K., Hawkes, T. (1992). Finite Soluble Groups. Berlin–New York: Walter de Gruyter, pp. 892] for notations and basic results.