Related papers: On some problems regarding $LCM$-groups

Denote the sum of element orders in a finite group $G$ by $\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. In this paper, we prove that if $|G|=n$ and $\psi(G)>\frac{13}{21}\,\psi(C_n)$, then $G$ is nilpotent. Moreover, we have…

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the…

Let $G$ be a finite group and $\sigma=\{\sigma_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}\sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}=\emptyset$ for all $i\neq j$. A chief factor…

Let $q$ be a prime and $A$ a finite $q$-group of exponent $q$ acting by automorphisms on a finite $q'$-group $G$. Assume that $A$ has order at least $q^3$. We show that if $\gamma_{\infty} (C_{G}(a))$ has order at most $m$ for any $a \in…

Let $o(G)$ be the average order of a finite group $G$. In this paper, we prove that if $o(G)<\frac{31}{12}$\,, then $G$ is supersolvable. Moreover, we have $o(G)=\frac{31}{12}$ if and only if $G\cong A_4$. We also classify finite groups $G$…

An $\mathfrak{M}_C$ group is a group in which all chains of centralizers have finite length. In this article, we show that every nilpotent subgroup of an $\mathfrak{M}_C$ group is contained in a definable subgroup which is nilpotent of the…

Let $q$ be a prime and $A$ an elementary abelian group of order at least $q^3$ acting by automorphisms on a finite $q'$-group $G$. It is proved that if $|\gamma_{\infty}(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of…

If $G$ is a finite group and $x\in G$ then the set of all elements of $G$ having the same order as $x$ is called {\em an order subset of $G$ determined by $x$} (see [2]). We say that $G$ is a {\em group with perfect order subsets} or…

It is well known that if $G$ is a group and $H$ is a normal subgroup of $G$ of finite index $k$, then $x^k \in H$ for every $x \in G$. We examine finite groups $G$ with the property that $x^k \in H$ for every subgroup $H$ of $G$, where $k$…

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every…

Let $G$ be a finite group and let $k \geq 2$. We prove that the coprime subgroup $\gamma_k^*(G)$ is nilpotent if and only if $|xy|=|x||y|$ for any $\gamma_k^*$-commutators $x,y \in G$ of coprime orders (Theorem A). Moreover, we show that…

We give a detailed description of infinite locally nilpotent groups G such that the index |C_G (x) : <x>| is finite, for every non-normal cyclic subgroup <x> of G. We are also able to extend our analysis to all non-periodic groups…

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…

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $\delta(G)$ and $\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \mathscr L $-$ \Pi $-property in $ G $ if $ H\unlhd G $, or if $ | G / K : \mathrm{N} _{G / K} (HK/K)| $ is a $ \pi (HK/K) $-number for any $ G…

We prove that if $G$ is a finite simple group which is the unit group of a ring, then $G$ is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order $2^k -1$ for some $k$; or (c) a projective special linear…

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is…

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 investigate the question of how many subgroups of a finite group are not in its Chermak-Delgado lattice. The Chermak-Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasol\u{a} and…