On 𝔛-Saturated Formations of Finite Groups

Abstract

In the paper, a Frattini-like subgroup associated with a class X of simple groups is introduced and analysed. The corresponding X-saturated formations are exactly the X-local ones introduced by Förster. Our techniques are also very useful to highlight the properties and behaviour of ω-local formations. In fact, extensions and improvements of several results of Shemetkov are natural consequences of our study.

$\mathfrak{F}$ -critical groups, $\mathfrak{F}$ -subnormal subgroups, and the generalised Wielandt property for residuals

“…If N is a minimal normal subgroup of G, it follows that G/N ∈ F. Therefore G is in the boundary of F and N = G F is the unique minimal normal subgroup of G. It is clear that N i = 1, i = 1, 2. Hence N is contained in N 1 ∩ N 2 and thus N i is F-subnormal in G, i = 1, 2 by Lemma 15 (1). Since F is a GWP-formation, it follows that G F = N F 1 N F 2 = 1, contrary to supposition.…”

“…This implies that HN is F-subnormal in G by Lemma 14 (2). Moreover HN ∈ E K(F) by Lemma 16 (1). Assume that HN is a proper subgroup of G. Since H is K-F-subnormal in HN by Lemma 15 (2), it follows that H is F-subnormal in HN by induction.…”

“…Except for a particular case, it is defined as Φ O X (G) , where O X (G) is the largest normal subgroup of G whose composition factors belong to X. It was proved in [1] that the new X-saturated formations also coincide with the X-local ones.…”

“…He proves that X-saturated formations are exactly the X-local ones. In [1], an alternative X-Frattini subgroup Φ X (G) of a group G is introduced. Except for a particular case, it is defined as Φ O X (G) , where O X (G) is the largest normal subgroup of G whose composition factors belong to X.…”

Products of formations of finite groups

“…He proves that X-saturated formations are exactly the X-local ones. In [4], an alternative X-Frattini subgroup Φ X (U ) of a group U is introduced. Except for a particular case, it is defined as Φ(O X (U )), where O X (U ) is the largest normal subgroup of U whose composition factors belong to X.…”