A new formation with Shemetkov property

Let F be a class of finite groups. A group G is called a minimal non-F-group or simply an F-critical group, if G is not in F but all proper subgroups of G are in F. Let Sch(G) denotes the set of all Schmidt subgroups of a group G (i.e. N-critical subgroups of G, where N is the class of all nilpotent groups), and SchU={G|Sch(G)⊆U}, where U is the class of all supersoluble groups. In the paper, we investigate the new properties of the class SchU. In particular, we prove that SchU is a subgroup-closed saturated Fitting formation with Shemetkov property, i.e. every SchU-critical group is a Schmidt group. In addition, we show that SchU is locally defined by the formation function f such that f(p)=S{p}∪π(p−1) for every prime p. Besides, we describe all SchU-critical groups and all minimal non-supersoluble groups that belong to SchU.