On $\mathfrak{F}_n$-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS

Let G be a finite group and denote a formation of finite groups by [Formula: see text]. We call a subgroup H of G[Formula: see text]-supplemented in G if there exists a normal subgroup K of G such that G = HK and (H ∩ K)HG/HG is contained in the [Formula: see text]-hypercenter [Formula: see text] of G/HG. In this paper, we use [Formula: see text]-supplemented subgroups to study the structure of finite groups. Some known results in the literature are generalized.

SCHREIER CONDITIONS ON CHIEF FACTORS AND RESIDUALS OF SOLVABLE-LIKE GROUP FORMATIONS

Let α be a formation of finite groups which is closed under subgroups and group extensions and which contains the formation of solvable groups. Let G be any finite group. We state and prove equivalences between conditions on chief factors of G and structural characterizations of the α-residual and theα-radical of G. We also discuss the connection of our results to the generalized Fitting subgroup of G.

Frattini classes of saturated formations of finite groups

We study the following question: given any local formation of finite groups, do there exist maximal local subformations? An answer is given by providing a local definition of the intersection of all maximal local subformations.

Local definitions of local homomorphs and formations of finite groups

It is well known that every local formation of finite soluble groups possesses three distinguished local definitions consisting of finite soluble groups: the minimal one, the full and integrated one, and the maximal one. As far as the first and the second of these are concerned, this statement remains true in the context of arbitrary finite groups. Doerk, Šemetkov, and Schmid have posed the problem of whether every local formation of finite groups has a distinguished (that is, unique) maximal local definition. In this paper a description of local formations with a unique maximal local definition is given, from which counter-examples emerge. Furthermore, a criterion for a formation function to be a local definition of a given local formation is obtained.