相关论文: On Maximal Subgroups of a Finite Solvable Group

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. We say that $H$ is s-semipermutable in $G$ if $HG_p = G_pH$ for any Sylow $p$-subgroup $G_p$ of $G$ with $(p, |H|) = 1$. We investigate the influence of s-semipermutable…

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…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. We provide a complete classification of a finite group $G$ in which every maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant…

We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$…

Let $H$ be a subgroup of a group $G$. The permutizer $P_G(H)$ is the subgroup generated by all cyclic subgroups of $G$ which permute with $H$. A subgroup $H$ of a group $G$ is strongly permutable in $G$ if $P_U(H)=U$ for every subgroup $U$…

Suppose that $G$ is a finite solvable group and let $H$ be a Hall $\pi$-subgroup, let $b(H)$ be the largest character degree of $H$, we show that $|G:O_{\pi' \pi}(G)|_{\pi} \leq b(H)^2$.

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of a finite group $G$. We prove that the finite groups with $\lambda(G)=|G|-t$, where $t\leq 5$, are solvable, and classify such groups.

Let $D$ be a division ring with center $F$, and $G$ an almost subnormal subgroup of $D^*$. In this paper, we show that if $G$ contains a non-abelian locally solvable maximal subgroup, then $D$ must be a cyclic algebra of prime degree over…

Let $G$ be a finite group, and $\pi$ be a set of primes. The $\pi$-core $\mathbf{O}_\pi(G)$ is the unique maximal normal $\pi$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved…

Let $G$ be a finite group. We let $\f{m}(G)$ and $\sig(G)$ denote the number of maximal subgroups of $G$ and the least positive integer $n$ such that $G$ is written as the union of $n$ proper subgroups, respectively. In this paper we…

A subgroup $H$ of a finite group $G$ is said to satisfy $\Pi$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $\Pi$-supplemented in…

Supersolubility of a finite group $G=\langle A,B\rangle$ with the nilpotent derived subgroup $G^\prime$ is established under the condition that the subgroups $A$ and $B$ are both subnormal and supersoluble.

In this paper, we determine the finite groups with a Sylow $r$-subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the $ \Pi $-property in $ G $ if for any chief factor $ L / K $ of $ G $, $ |G/K : N_{G/K}(HK/K\cap L/K )| $ is a $ \pi (HK/K\cap L/K) $-number. In this paper, we…

We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying…

Let $G$ be a finite solvable group and $H$ a non-normal core-free subgroup of $G$. We show that if the normalizer of any non-trivial normal subgroup of $Fit(H)$ is equal $H$, then $H$ has a nilpotent normal complement $K$ such that $G=KH$…

Let $G$ be a finite group and let $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. We conjecture that there exists $x \in G$ such that $P_i \cap P_i^x$ is inclusion-minimal in $\{ P_i \cap P_i^g : g \in G\}$ for all…

Let $M$ be a maximal subgroup of a finite group $G$ and $K/L$ be a chief factor such that $L\leq M$ while $K\nsubseteq M$. We call the group $M\cap K/L$ a $c$\ns section of $M$. And we define $Sec(M)$ to be the abstract group that is…

Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) \leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, we first prove some results on the solvability of finite groups in which some maximal $A$-invariant subgroups have indices a prime or the square of a…