K. Doerk和P. Hall分别给出的两个经典结果，确立了所有极大子群均为超可解的有限群的可解性，以及所有极大子群指数为素数或素数平方的有限群的可解性。本文描述了极大子群要么是超可解、要么指数为素数或素数平方的非可解有限群的结构。
英文摘要：
Two classic results, due to K. Doerk and P. Hall respectively, establish the solvability of those finite groups all of whose maximal subgroups are supersolvable, and the solvability of finite groups in which all maximal subgroups have prime or squared prime index. In this note we describe the structure of the non-solvable finite groups whose maximal subgroups are either supersolvable or have prime or squared prime index.
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.
We study finite groups $G$ with the property that for any subgroup $M$ maximal in $G$ whose order is divisible by all the prime divisors of $|G|$, $M$ is supersolvable. We show that any nonabelian simple group can occur as a composition factor of such a group and that, if $G$ is solvable, then the nilpotency length and the rank are arbitrarily large. On the other hand, for every prime $p$, the $p$-length of such a group is at most $1$. This answers questions proposed by V. Monakhov in The Kourovka Notebook.
Let $p$ be a prime number, $G$ be a $p$-solvable finite group and $P$ be a Sylow $p$-subgroup of $G$. We prove that $G$ is $p$-supersolvable if $N_G(P)$ is $p$-supersolvable and if there is a subgroup $H$ of $P$ with $P' \le H \le Φ(P)$ such that $H$ is $s$-semipermutable in $G$. As applications, we simplify the proofs of some known results and also generalize some known results.
Let $\mathscr{F}$ be a formation and $G$ a finite group. The weak norm of a subgroup $H$ in $G$ with respect to $\mathscr{F}$ is defined by $N_{\mathscr{F}}(G,H)=\underset{T\leq H}{\bigcap}N_G(T^{\mathscr{F}})$. In particular, $N_{\mathscr{F}}(G)=N_{\mathscr{F}}(G,G)$. Let $N^i_{\mathscr{F}}(G)$,$i\geq 1$, be a upper series of $G$ by setting $N^0_{\mathscr{F}}(G)=1$, $N^{i+1}_{\mathscr{F}}(G)/N^i_{\mathscr{F}}(G)=N_{\mathscr{F}}(G/N^i_{\mathscr{F}}(G))$ and denoted by $N^{\infty}_{\mathscr{F}}(G)$ the terminal term of the series. In this paper, for the case $\mathscr{F}\in\{\mathscr{U}_p,\math…
A subgroup $H$ of a group $G$ is said to be an $ICΦ$-subgroup of $G$ if $H \cap [H,G] \le Φ(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $ICΦ$-subgroups of $G$. A new characterization of finite abelian groups and some new criteria for $2$-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.
Let $G$ be a finite group and $p^k$ be a prime power dividing $|G|$. A subgroup $H$ of $G$ is called to be $\mathcal{M}$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H_iK<G$ for every maximal subgroup $H_i$ of $H$. In this paper, we complete the classification of the finite groups $G$ in which all subgroups of order $p^k$ are $\mathcal{M}$-supplemented. In particular, we show that if $k\geq 2$, then $G/\mathrm{O}_{p'}(G)$ is supersolvable with a normal Sylow $p$-subgroup and a cyclic $p$-complement.
Let $G$ be a finite group and $σ_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $σ_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_4$ and $A_4$ are obtained. We also show that $σ_1(G)<c$ does not imply the supersolvability of $G$ for no constant $c\in(2,\infty)$.
Let $G$ be a group and $H \le K \le G$. We say that $H$ is $c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H \cap B \le Z(K)$. Given a finite group $G$, a prime number $p$ and a Sylow $p$-subgroup $P$ of $G$, we investigate the structure of $G$ under the assumption that $N_G(P)$ is $p$-supersolvable or $p$-nilpotent and that certain cyclic subgroups of $P$ with order $p$ or $4$ are $c$-embedded in $G$ with respect to $P$. New characterizations of $p$-supersolvability and $p$-nilpotence of finite groups will be obtained.
Let $λ(G)$ be the maximum number of subgroups in an irredundant covering of the finite group $G$. We prove that if $G$ is a group with $λ(G) \leqslant 6$, then $G$ is supersolvable. We also describe the structure of the groups $G$ with $λ(G)=6$. Moreover, we show that if $G$ is a group with $λ(G) \leqslant 30$, then $G$ is solvable.
Let $k$ be any field and let $G$ be a connected reductive algebraic $k$-group. Associated to $G$ is an invariant first studied by Satake and Tits that is called the index of $G$ (a Dynkin diagram along with some additional combinatorial information). Tits showed that the $k$-isogeny class of $G$ is uniquely determined by its index and the $k$-isogeny class of its anisotropic kernel $G_a$. For the cases where $G$ is absolutely simple, Satake and Tits classified all possibilities for the index of $G$. Let $H$ be a connected reductive $k$-subgroup of maximal rank in $G$. We introduce an invariant…
Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. We prove that $p$ does not divide the conjugacy class size (index) of each $p$-regular element of prime power order $x\in A\cup B$ if and only if $G$ is $p$-decomposable, i.e. $G=O_p(G) \times O_{p'}(G)$.
In this paper we define and investigate a class of groups characterized by a representation-theoretic property we call purely noncommuting or PNC. This property guarantees that the group has an action on a smooth projective variety with mild quotient singularities. It has intrinsic group-theoretic interest as well. The main results are as follows. (i) All supersolvable groups are PNC. (ii) No nonabelian finite simple groups are PNC. (iii) A metabelian group is guaranteed to be PNC if its commutator subgroup's cyclic prime-power-order factors are all distinct, but not in general. We also give a…
Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$ whose elements are the conjugacy class sizes of the vanishing elements of $G$. Motivated by the results in \cite{BLP}, we describe the class of the finite groups $G$ such that vcs$(G)$ consists of a single element \emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylo…
On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow $2$-subgroup of AGL(2,n), each term of the chain is defined as the normalizer of the previous one in the symmetric group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, for large values of $n$, the index of a normalizer in the consecutive one does not depend on $n$. Indeed, there is a strong evidence that the sequence of the logarithms of such indices is the one of the partial sums of the numbers of partitions into a…
We describe an algorithm for computing Schur indices of irreducible characters of a finite group $G$, based on computations within $G$ and its subgroups and with their character tables. The algorithm has been implemented within \Magma\ and examples of its use are given. We also compare the output of the algorithm on groups associated with sporadic simple groups with the results of W. Feit. The differences are considered.
We prove that the Herzog-Schönheim Conjecture holds for any group $G$ of order smaller than $1440$. In other words we show that in any non-trivial coset partition $\{g_i U_i\}_{i=1}^n $ of $G$ there exist distinct $1 \leq i, j \leq n$ such that $[G:U_i]=[G:U_j]$. We also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if $U_1$,...,$U_n$ are subgroups of $G$ which have pairwise trivially intersecting cosets and $n \leq 4$ then $[G:U_1]$,...,$[G:U_n]$ are harmonic integers.
Given a GGS-group $G$ with non-constant defining tuple over a prime-regular rooted tree, we calculate the indices $|G:G^{(n)}|$ and describe the structure of the higher derived subgroups $G^{(n)}$ for all $n \in \mathbb{N}$. We find that the values $|G:G^{(n)}|$ depend only mildly on the structure of the defining tuple.
