On finite minimal non-p-supersoluble groups
If ℱ is a class of groups, then a minimal non-ℱ-group (a dual minimal non-ℱ-group resp.) is a group which is not in ℱ but any of its proper subgroups (factor groups resp.) is in ℱ. In many problems of classification of groups it is sometimes useful to know structure properties of classes of minimal non-ℱ-groups and dual minimal non-ℱ-groups. In fact, the literature on group theory contains many results directed to classify some of the most remarkable among the aforesaid classes. In particular, V. N. Semenchuk in  and  examined the structure of minimal non-ℱ-groups for ℱ a formation, proving, among other results, that if ℱ is a saturated formation, then the structure of finite soluble, minimal non-ℱ-groups can be determined provided that the structure of finite soluble, minimal non-ℱ-groups with trivial Frattini subgroup is known. In this paper we use this result with regard to the formation of p-supersoluble groups (p prime), starting from the classification of finite soluble, minimal non-p-supersoluble groups with trivial Frattini subgroup given by N. P. Kontorovich and V. P. Nagrebetskiĭ (). The second part of this paper deals with non-soluble, minimal non-p-supersoluble finite groups. The problem is reduced to the case of simple groups. We classify the simple, minimal non-p-supersoluble groups, p being the smallest odd prime divisor of the group order, and provide a characterization of minimal simple groups. All the groups considered are finite.