相关论文: Finite groups with a certain number of elements pa…

For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le n}. If s is a positive integer and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S of G…

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…

A classical result of Baer states that a finite group $ G $ which is the product of two normal supersoluble subgroups is supersoluble if and only if $ G' $ is nilpotent. In this article we show that if $ G=AB $ is the product of…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. For any fixed prime divisor $p$ of $|G|$, we provide a complete characterization of the structure of a group $G$ in which every maximal $A$-invariant…

Suppose that $\mathcal{C}$ is a class of groups consisting only of periodic groups and $\mathfrak{P}(\mathcal{C})^{\prime}$ is the set of prime numbers each of which does not divide the order of any element of a $\mathcal{C}$-group. A…

Let $G$ be a group. A subset $D$ of $G$ is a determining set of $G$, if every automorphism of $G$ is uniquely determined by its action on $D$. The determining number of $G$, denoted by $\alpha(G)$, is the cardinality of a smallest…

Consider a nonsolvable finite group G, where R(G) represents the solvable radical of G. For any element x in G, the solvabilizer of x in G, denoted by Sol_G(x), is defined as the set of all elements y in G such that the subgroup generated…

A group $G$ is integrable if it is isomorphic to the derived subgroup of a group $H$; that is, if $H'\simeq G$, and in this case $H$ is an integral of $G$. If $G$ is a subgroup of $U$, we say that $G$ is integrable within $U$ if $G=H'$ for…

A finite group $G$ is said to have the nilpotent decomposition property (ND) if for every nilpotent element $\alpha$ of the integral group ring $\mathbb{Z}[G]$ one has that $\alpha e$ also belong to $\mathbb{Z}[G]$, for every primitive…

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…

Let $G$ be a finite group and denote by $o(g)$ the order of an element $g\in G$. We say that $G$ is an $LCM$-group if $o(x^ny)$ is a divisor of the least common multiple of $o(x^n)$ and $o(y)$ for all $x, y\in G$ and $n\in\mathbb{N}$. This…

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set $\Bbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $\sigma$-central if the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is a…

A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xy\in\{x,y\}$ ($xy\notin\{x,y\}$) for any (distinct) elements $x,y\in S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We…

Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\dots,x_{k+1}$ satisfy $[x_1,x_2,\dots,x_{k+1}]=1$. We study the following problem: If…

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…

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing the element $x$. A famous theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group…

Let $G$ be a finite solvable group and $H$ a non-normal core-free subgroup of $G$. We show that if the normalizer of any non-trivial normal subgroup of $Fit(H)$ is equal $H$, then $H$ has a nilpotent normal complement $K$ such that $G=KH$…

Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such…

Let $m,n$ be positive integers and $p$ a prime. We denote by $\nu(G)$ an extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is a residually finite group satisfying some non-trivial identity $f…