On finite minimal non-nilpotent groups

Abstract

A critical group for a class of groups X \mathfrak {X} is a minimal non- X \mathfrak {X} -group. The critical groups are determined for various classes of finite groups. As a consequence, a classification of the minimal non-nilpotent groups (also called Schmidt groups) is given, together with a complete proof of Gol’fand’s theorem on maximal Schmidt groups.