Some generalizations of Shao and Beltrán’s theorem

Let [Formula: see text] and [Formula: see text] be finite groups of relative coprime orders and [Formula: see text] act on [Formula: see text] via automorphisms. In this paper, we prove that when every maximal [Formula: see text]-invariant subgroup of [Formula: see text] that contains the normalizer of some Sylow subgroup has prime index, then [Formula: see text] is supersolvable; if every non-nilpotent maximal [Formula: see text]-invariant subgroup of [Formula: see text] has prime index or is normal in [Formula: see text], then [Formula: see text] is a Sylow tower group.

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.

Sylow towers in groups where the index of every non-nilpotent maximal subgroup is prime

In this paper, we prove that if every non-nilpotent maximal subgroup of a finite group [Formula: see text] has prime index then [Formula: see text] has a Sylow tower.

Coprime actions with supersolvable fixed-point groups

Let A be an elementary abelian r-group acting on a finite {r^{\prime}} -group G. Suppose that the fixed-point group {\operatorname{C}_{G}(a)} is supersolvable for each {a\in A^{\#}} . We show that G is supersolvable if {|A|\geqslant r^{4}} and that {G^{\prime}\leqslant\mathbf{F}_{3}(G)} if {|A|\geqslant r^{3}} . Moreover, we prove some other results for cases when the fixed-point group {\operatorname{C}_{G}(a)} is abelian, p-nilpotent or satisfies the Sylow tower property.

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 $H\cap N=1$.

GROUPS OF FINITE NORMAL LENGTH

Let $k$ be a nonnegative integer. A subgroup $X$ of a group $G$ has normal length $k$ in $G$ if all chains between $X$ and its normal closure $X^{G}$ have length at most $k$, and $k$ is the length of at least one of these chains. The group $G$ is said to have finite normal length if there is a finite upper bound for the normal lengths of its subgroups. The aim of this paper is to study groups of finite normal length. Among other results, it is proved that if all subgroups of a locally (soluble-by-finite) group $G$ have finite normal length in $G$, then the commutator subgroup $G^{\prime }$ is finite and so $G$ has finite normal length. Special attention is given to the structure of groups of normal length $2$. In particular, it is shown that finite groups with this property admit a Sylow tower.

Languages associated with saturated formations of groups

In a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.