A new formation with Shemetkov property

Abstract

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 Sch U = { G | Sch ( G ) ⊆ U } , where U is the class of all supersoluble groups. In the paper, we investigate the new properties of the class Sch U . In particular, we prove that Sch U is a subgroup-closed saturated Fitting formation with Shemetkov property, i.e. every Sch U -critical group is a Schmidt group. In addition, we show that Sch U 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 Sch U -critical groups and all minimal non-supersoluble groups that belong to Sch U .