Related papers: Criteria for solvable radical membership via p-ele…

Let $\Bbb P$ be the set of all primes. A subgroup $H$ of a group $G$ is called {\it $\mathbb P$-subnormal} in $G$, if either $H=G$, or there exists a chain of subgroups $H=H_0\le H_1\le \ldots \le H_n=G, \ |H_{i}:H_{i-1}|\in \Bbb P, \…

A subgroup $H$ of a finite group $G$ is said to be an NC-subgroup of $G$, if $ H^G N_G (H) =G$, where $H^G$ denotes the normal closure of $H$ in $G$. A finite group $G$ is called a PNC-group, if any subgroup of $G$ is an NC-subgroup of $G$,…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $\sigma_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $\sigma_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$.…

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of…

Fix an odd prime $p$, and let $F$ be a field containing a primitive $p$th root of unity. It is known that a $p$-rigid field $F$ is characterized by the property that the Galois group $G_F(p)$ of the maximal $p$-extension $F(p)/F$ is a…

Suppose that a locally finite group $G$ has a $2$-element $g$ with Chernikov centralizer. It is proved that if the involution in $\langle g\rangle$ has nilpotent centralizer, then $G$ has a soluble subgroup of finite index.

Let $G$ be a finite insoluble group with soluble radical $ R(G)$. The solubility graph $\Gamma_{\rm S}(G)$ of $G$ is a simple graph whose vertices are the elements of $G\setminus R(G) $ and two distinct vertices $x$ and $y$ are adjacent if…

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…

Let $G$ be a finite group, and assume that $G$ has an automorphism of order at least $\rho|G|$, with $\rho\in\left(0,1\right)$. Generalizing recent analogous results of the author on finite groups with a large automorphism cycle length, we…

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of…

We investigate the structure of finite groups whose non-central real class sizes have the same $2$-part. In particular, we prove that such groups are solvable and have $2$-length one. As a consequence, we show that a finite group is…

We prove that each \omega-categorical, generically stable group is solvable-by-finite.

The prime graph of a finite group $G$ is the labelled graph $\Gamma(G)$ with vertices the prime divisors of $|G|$ and edges the pairs $\{p,q\}$ for which $G$ contains an element of order $pq$. A group $G$ is recognisable by its prime graph…

If V is a finitely generated variety such that the first-order theory of the finite members of V is decidable, we show that V is residually finite, and in fact has a finite bound on the sizes of subdirectly irreducible algebras. This result…

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

Let $G$ be a finite group, let $p$ be a prime and let $w$ be a group-word. We say that $G$ satisfies $P(w,p)$ if the prime $p$ divides the order of $xy$ for every $w$-value $x$ in $G$ of $p'$-order and for every non-trivial $w$-value $y$ in…

We exhibit infinitely many natural numbers $n$ for which there exists at least one insolvable group of order $n$, and yet the holomorph of any solvable group of order $n$ has no insolvable regular subgroup. We also solve Problem 19.90 (d)…

We classify all finite groups $G$ which possesses an element $x\in G$ such that every irreducible character of $G$ takes a root of unity value at $x$.

The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L-series suggest a generalization of these groups, namely to such groups where every irreducible character has some multiple which is…