Related papers: Normal restriction in finite groups

Over any partially ordered abelian group whose positive cone is closed in an appropriate sense and has finitely many faces, modules that satisfy a weak finiteness condition admit finite primary decompositions. This conclusion rests on the…

We prove that Hall subgroups of finite simple groups are pronormal. Thus we obtain an affirmative answer to Problem 17.45(a) of "Kourovka notebook".

In the spirit of an earlier result of M\"uller on the Heisenberg group we prove a restriction theorem on a certain class of two step nilpotent Lie groups. Our result extends that of M\"uller also in the framework of the Heisenberg group.

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant…

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without…

A characterization of congruences in free semigroups is presented.

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper we classify these rings and show that all finite rings of order $p^n$ for $n < 5$ and some of…

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other.…

We prove a version of Hrushovski's socle lemma for rigid groups in an arbitrary simple theory.

We show that a finite group $G$ admitting an automorphism $\alpha$ such that the function $G\rightarrow G$, $g\mapsto g\alpha(g)$, is bijective is necessarily solvable.

Computations in the cohomology of finite groups.

Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that…

In this report we summarize this work, all finite simple groups $G$ can determined uniformly using their orders $|G|$ and the set $\pi_e(G)$ of their element orders.

Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega(G),$ where $\Omega(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this…

Let $k$ be a field of characteristic $p>0$, which has infinitely many discrete valuations. We show that every finite embedding problem for $\Gal(k)$ with finitely many prescribed local conditions, whose kernel is a $p$-group, is properly…

In this paper, we classify the finite simple groups with an abelian Sylow subgroup.

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…

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble…

The aim of this short note is to provide a proof of the decidability of the generalized membership problem for relatively quasi-convex subgroups of finitely presented relatively hyperbolic groups, under some reasonably mild conditions on…

We prove the following: For any $\rho\in\left(0,1\right)$, if a finite group $G$ has an automorphism with a cycle of length at least $\rho\cdot|G|$, then the index of the solvable radical $\operatorname{Rad}(G)$ in $G$ is bounded from above…