On NH-embedded subgroups of finite groups

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. [Formula: see text] is said to be [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a Hall subgroup of [Formula: see text] and [Formula: see text], where [Formula: see text] is the largest [Formula: see text]-semipermutable subgroup of [Formula: see text] contained in [Formula: see text]. In this paper, we give some new characterizations of [Formula: see text]-nilpotent and supersolvable groups by using [Formula: see text]-embedded subgroups. Some known results are generalized.

On normal subgroups of M-groups

The normal embedding problem of finite solvable groups into [Formula: see text]-groups was studied. It was proved that for a finite solvable group [Formula: see text], if [Formula: see text] has a special normal nilpotent Hall subgroup, then [Formula: see text] cannot be a normal subgroup of any [Formula: see text]-group; on the other hand, if [Formula: see text] has a maximal normal subgroup which is an [Formula: see text]-group, then [Formula: see text] can occur as a normal subgroup of an [Formula: see text]-group under some suitable conditions. The results generalize the normal embedding theorem on solvable minimal non-[Formula: see text]-groups to arbitrary [Formula: see text]-groups due to van der Waall, and also cover the famous counterexample given by Dade and van der Waall independently to the Dornhoff’s conjecture which states that normal subgroups of arbitrary [Formula: see text]-groups must be [Formula: see text]-groups.

On the First Zassenhaus Conjecture and Direct Products

In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups.(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G\times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

A NOTE ON NORMAL COMPLEMENTS FOR FINITE GROUPS

Assume that $G$ is a finite group and $H$ is a 2-nilpotent Sylow tower Hall subgroup of $G$ such that if $x$ and $y$ are $G$-conjugate elements of $H\cap G^{\prime }$ of prime order or order 4, then $x$ and $y$ are $H$-conjugate. We prove that there exists a normal subgroup $N$ of $G$ such that $G=HN$ and $H\cap N=1$.

𝔉-groups and Hall s-semiembedded subgroups

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be a Hall [Formula: see text]-semiembedded subgroup of [Formula: see text] if [Formula: see text] is a Hall subgroup of [Formula: see text] for any [Formula: see text], where [Formula: see text]. In this paper, we investigate the influence of Hall [Formula: see text]-semiembedded subgroups on the structure of the finite group [Formula: see text]. Some new results about [Formula: see text] to be a [Formula: see text]-group are obtained, where [Formula: see text] is a saturated formation.