Related papers: On CLT and non-CLT groups

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

It is shown that finite groups in which the order of the product of every pair of elements of co-prime order is the product of the orders, is nilpotent.

Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let \rho(G) be the set of all primes which divide some character degree of G. The prime graph \Delta(G) attached to G is a graph whose vertex…

For every non-nilpotent finite group $G$, there exists at least one proper subgroup $M$ such that $G$ is the setwise product of a finite number of conjugates of $M$. We define $\gamma_{\text{cp}}\left( G\right) $ to be the smallest number…

It is proved that the derived subgroup of a finite group is nilpotent if and only if $|ab|\ge |a||b|$ for all primary commutators $a$ and $b$ of coprime orders.

Let $G$ be a finite group and $n_p(G)$ the number of Sylow $p$-subgroups of $G$. In this paper, we prove if $n_p(G)<p^2$ then almost all numbers $n_p(G)$ are a power of a prime.

Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2, \ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<\cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${\rm GK}(G)$, is defined as follows: its vertex set is…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \Pi $-property in $ G $ if there exists a $G$-chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that $ | G…

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with…

Given a finite group G with an irreducible character \chi \in Irr(G), the codegree of \chi is defined by cod(\chi) = |G :\ker \chi|/\chi(1). The set of non-linear irreducible character codegrees of G is denoted by cod(G|G'). In this note,…

Let $G$ be a finite $p$-group and $\delta(G)$ denote the number of all non-cyclic subgroups of $G$. In this paper, an upper bound for $\delta(G)$ is obtained. Furthermore, we prove that $\delta(G)\leq \delta(M_p(1, 1, 1) \times…

Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\{g\in G \ | \ g^{p^n}=1\}$ is a nontrivial subgroup for some $n$, then $G$ is a…

Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $\delta_k$-values $a,b\in G$ of coprime orders. In the course of the…

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 non-abelian group and $m=|G|/|Z(G)|$. In this paper we investigate $m$-centralizer group $G$ with cyclic center and we will prove that if $G$ is a finite non-abelian $m$-centralizer $CA$-group, then there exists an…

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…

Given a finite group G, let cd(G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd(G),…

In this paper we investigate the following general problem. Let $G$ be a group and let $i(G)$ be a property of $G$. Is there an integer $d$ such that $G$ contains a $d$-generated subgroup $H$ with $i(H)=i(G)$? Here we consider the case…

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is…

In this note, we prove: \medskip \noindent {\bf Theorem A:} \emph{ There is a fixed constant $C$ such that for any positive integer $n$ and prime $p$, every finite subgroup $G$ of order coprime to $p$ of ${\rm GL}(n,\mathbb{C})$ has an…