Related papers: On a group-theoretical generalization of the Euler…

Let $G$ be a finite group, $A$ a finite abelian group. Each homomorphism $\phi:G\to A\wr S_n$ induces a homomorphism $\bar{\phi}:G\to A$ in a natural way. We show that as $\phi$ is chosen randomly, then the distribution of $\bar{\phi}$ is…

We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non--cyclic free group, and $g$ is an element of $H$, then the…

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…

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is a finitely generated group in which the set of all simple tensors $T_{\otimes}(G)$ is…

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 $o(G)$ 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 $G$…

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $\varphi(q,p(x))$ be the Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$ is a finite…

Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a…

Given a positive integer $u$ and a simple algebraic group $G$ defined over an algebraically closed field $K$ of characteristic $p$, we derive properties about the subvariety $G_{[u]}$ of $G$ consisting of elements of $G$ of order dividing…

For a variety of finite groups $\mathbf H$, let $\overline{\mathbf H}$ denote the variety of finite semigroups all of whose subgroups lie in $\mathbf H$. We give a characterization of the subsets of a finite semigroup that are pointlike…

Let $n$ be a positive integer. Then cyclic group $Z_n$ of order $n$ is the only group of order $n$ iff g.c.d. $(n,\phi(n))=1$, where $\phi$ denotes the Euler-phi function. In this article we have given another proof of this result using the…

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…

By a result known as Rieger's theorem (1956), there is a one-to-one correspondence, assigning to each cyclically ordered group $H$ a pair $(G,z)$ where $G$ is a totally ordered group and $z$ is an element in the center of $G$, generating a…

We define a class $\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\mathcal{U}$ and…

Let $G$ be a group of order $n$ and $H$ be a subgroup of order $m$ of $G$. Denote by $\psi_H(G)$ the sum of element orders relative to $H$ of $G$. It is known that if $G$ is nilpotent, then $\psi_H(G)\leq\psi_{H_m}(G)$, where $H_m$ is the…

Suppose $\ell$ is a prime number, $\ell >3$, $K$ is a field that is an unramified finite extension of the field $\Q_\ell$ of $\ell$-adic numbers, and $G$ is a finite group that is a semi-direct product of a normal $\ell'$-subgroup $H$ and a…

The Euler's totient function $ \varphi(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ \varphi(n)\mid n-1…

Let $G$ be a group. We give an explicit description of the set of elements $x \in G$ such that $x^{|G:H|} \in H$ for every subgroup of finite index $H \leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$,…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$…

Let $G$ be any group. The quotient group $T(G)$ of the multiple holomorph by the holomorph of $G$ has been investigated for various families of groups $G$. In this paper, we shall take $G$ to be a finite $p$-group of class two for any odd…