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Register a new userA $G$-covering subgroup system of a finite group for some classes of $sigma$-soluble groups
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Let ${frak F}$ be a class of group and $G$ a finite group. Then a set $Sigma $ of subgroups of $G$ is called a emph{$G$-covering subgroup system} for the class ${frak F}$ if $Gin {frak F}$ whenever $Sigma subseteq {frak F}$. We prove that: {sl If a set of subgroups $Sigma$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $Sigma$ is a $G$-covering subgroup system for the classes of all $sigma$-soluble and all $sigma$-nilpotent groups, and for the class of all $sigma$-soluble $Psigma T$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook.
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Let $sigma ={sigma_{i} | iin I}$ be a partition of the set of all primes $Bbb{P}$ and $G$ a finite group. Let $sigma (G)={sigma _{i} : sigma _{i}cap pi (G) e emptyset$. A set ${cal H}$ of subgroups of $G$ is said to be a complete Hall $sigma $-set of $G$ if every member $ e 1$ of ${cal H}$ is a Hall $sigma _{i}$-subgroup of $G$ for some $iin I$ and $cal H$ contains exactly one Hall $sigma _{i}$-subgroup of $G$ for every $i$ such that $sigma _{i}in sigma (G)$. We say that $G$ is $sigma$-full if $G$ possesses a complete Hall $sigma $-set. A complete Hall $sigma $-set $cal H$ of $G$ is said to be a $sigma$-basis of $G$ if every two subgroups $A, B incal H$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $sigma$-basis. In particular, we prove that if $G$ has a a $sigma$-basis, then $G$ is generalized $sigma$-soluble, that is, $G$ has a complete Hall $sigma $-set and for every chief factor $H/K$ of $G$ we have $|sigma (H/K)|leq 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281--309], we prove the following Theorem A. Suppose that $G$ is $sigma$-full. Then every complete Hall $sigma$-set of $G$ forms a $sigma$-basis of $G$ if and only if $G$ is generalized $sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $sigma (H/K)=sigma (G/C_{G}(H/K))$ or $sigma (H/K) ={sigma _{i}}$ and $G/C_{G}(H/K)$ is a $sigma _{i} cup sigma _{j}$-group for some $i e j$.
In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an element of order $pq$ in $G$. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any $nin mathbb{N}$, solvable groups of $n^text{th}$-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or $A_5$ and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.
Let $mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${cal L}_{mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}in mathfrak{F}$. A chief factor $H/K$ of $G$ is $mathfrak{F}$-central in $G$ if $(H/K)rtimes (G/C_{G}(H/K)) inmathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $mathfrak{F}$-central in $G$ for every subgroup $Ain {cal L}_{mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}leq Z_{infty}(G/A_{G})$ for every subgroup $Ain {cal L}_{mathfrak{N}}(G)$, where $mathfrak{N}$ is the class of all nilpotent groups.
We study the class of finite groups $G$ satisfying $Phi (G/N)= Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $delta_k$-values $a,bin G$ of coprime orders. In the course of the proof we establish the following result of independent interest: Let $P$ be a Sylow $p$-subgroup of $G$. Then $Pcap G^{(k)}$ is generated by $delta_k$-values contained in $P$. This is related to the so-called Focal Subgroup Theorem.
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