Related papers: Groups with normal restriction property

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…

It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group $G$ the subgroup $\gamma_{k}(G)$ is…

Let (G, *) be a semigroup, D subset of G, and n >= 2 be an integer. We say that (D, *) is an n-closed subset of G if a_1* ... *a_n in D for every a_1, ..., a_n in D. Hence every closed set is a 2-closed set. The concept of n-closed sets…

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

A group $G$ is said to have dense normalizers if each non-empty open interval in its subgroup lattice $L(G)$ contains the normalizer of a certain subgroup of $G$. In this note, we find all finite groups satisfying this property. We also…

Let $\sigma =\{\sigma_i |i\in I\}$ is some partition of all primes $\mathbb{P}$ and $G$ a finite group. A subgroup $H$ of $G$ is said to be $\sigma$-subnormal in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$…

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 study the groups $G$ with the curious property that there exists an element $k\in G$ and a function $f\colon G\to G$ such that $f(xk)=xf(x)$ holds for all $x\in G$. This property arose from the study of near-rings and input-output…

Let $G$ be a finite group and $p$ a fixed prime divisor of $|G|$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p'$-order, then (1)…

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 G be a group and H be a subgroup of G which is either finite or of finite index in G. In this note, we give some characterizations for normality of H in G. As a consequence we get a very short and elementary proof of the Main Theorem of…

Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the proper subgroups of $G$ and in which two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle.$ We prove that if there exists a…

In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially…

A subgroup $H$ of a group $G$ is said to be {pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. Some problems in finite group theory, combinatorics, and permutation group theory were solved in…

The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the…

A group $G$ is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of $G$, there exists a finite quotient of $G$ where the images of these subgroups are not conjugate. We prove that limit…

A $k$-tuple $(H_1, \ldots, H_k)$ of core-free subgroups of a finite group $G$ is said to be regular if $G$ has a regular orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_k$. The regularity number of $G$, denoted $R(G)$, is the…

A subgroup $A$ of a finite group $G$ is said to be a $CAP$-subgroup of $G$, if for any chief factor $H/K$ of $G$, either $A H= AK$ or $A\cap H = A \cap K$. Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion…

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of the finite group $G$. We prove that if $G$ is a group with $\lambda(G) \leqslant 6$, then $G$ is supersolvable. We also describe the structure of the groups…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall…