Related papers: Existence criterion for Hall subgroups of finite g…

In the paper new criteria of existence and conjugacy of Hall subgroups of finite groups are given.

In the paper we obtain the existence criterion of a Carter subgroup in a finite group in terms of its normal series. An example showing that the criterion cannot be reformulated in terms of composition factors is given.

A finite group $G$ is said to satisfy $C_\pi$ for a set of primes $\pi$, if $G$ possesses exactly one class of conjugate $\pi$-Hall subgroups. In the paper we obtain a criterion for a finite group $G$ to satisfy $C_\pi$ in terms of a normal…

Let $G$ be a finite group and let $\pi$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $\pi$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $\pi$-Hall subgroup if, and only if, $G$ has…

Let $G$ be a finite group. In this short note, we give a criterion of nilpotency of $G$ based on the existence of elements of certain order in each section of $G$.

In this note we introduce and characterize a class of finite groups for which the element orders satisfy a certain inequality. This is contained in some well-known classes of finite groups.

A subgroup $H$ of a finite group $G$ is said to be an $\mathscr{H}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g \cap N_T(H)\leq H$ for all $g\in G$. In this paper, we investigate the structure of…

We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups which can easily be verified from the character table.

A subgroup $H$ of a group $G$ is said to be an $IC\Phi$-subgroup of $G$ if $H \cap [H,G] \le \Phi(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $IC\Phi$-subgroups of $G$. A new…

In this paper we find the number of conjugate $\pi$-Hall subgroups in all finite almost simple groups. We also complete the classification of $\pi$-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.

In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G)$ holds. The…

Necessary and sufficient conditions for finite semihypergroups to be built from groups of the same order are established

Let $\mathbb {F}_q$ be a finite field and $G$ a finte group with $(|G|,q)=1$. By a group code in $\mathbb {F}_q[G]$ we mean a two-sided ideal in $\mathbb {F}_q[G]$. We will prove a general criterion for the existence of group codes with…

We give a criterion for an HNN extension of a finite $p$-group to be residually $p$.

In this article we introduce and study a class of finite groups for which the orders of normal subgroups satisfy a certain inequality. It is closely connected to some well-known arithmetic classes of natural numbers.

In this paper, we study a group in which every 2-maximal subgroup is a Hall subgroup.

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 this paper, we will prove some sufficient conditions for the solvability of groups.

Let $G$ be a finite soluble group and $h(G)$ its Fitting length. The aim of this paper is to give certain upper bounds for $h(G)$ as functions of the Fitting length of at least three Hall subgroups of $G$ which factorize $G$ in a particular…

In the paper, it is proved that if a finite group $G$ possesses a $\pi$-Hall subgroup for a set $\pi$ of primes, then every normal subgroup $A$ of $G$ possesses a $\pi$-Hall subgroup $H$ such that ${G=AN_G(H)}$.