A critical group for a class of groups $\mathfrak {X}$ is a minimal non- $\mathfrak {X}$ -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.

The aim of this paper is to classify the finite minimal non- p -supersoluble groups, p a prime number, in the p -soluble universe.

Characterisations of finite groups in which normality is a transitive relation are presented in the paper. We also characterise the finite groups in which every subgroup is either permutable or coincides with its permutiser as the groups in which every subgroup is permutable.