Related papers: On direct product subgroups of $\mathrm{SO}_3(\mat…

We address the problem of determining the class of self-similar groups, and in particular its closure under restricted direct products. We show that the group $\mathbb Z^{(\omega)}$ is self-similar, that $G^{(\omega)}\rtimes C_2$ is…

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur…

Let $G_{2}$ be a group which acts trivially on an abelian group $G_{1}$. As is well known, each perturbed direct product of $G_{1}$ and $G_{2}$ under a 2-cocycle $\varepsilon\in Z^{2}(G_{2},G_{1})$ determines a central extension of $G_{1}$…

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 finitely generated group. We prove that the $n$-fold tensor product $G^{\otimes n}$ is finite (resp. polycyclic) if and only $G$ is finite (resp. polycyclic). Further, assuming that $G$ is finitely presented, we show that…

Chevalley's theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian…

We provide lower bounds on the number of subgroups of a group $G$ as a function of the primes and exponents appearing in the prime factorization of $|G|$. Using these bounds, we classify all abelian groups with 22 or fewer subgroups, and…

We call a subfactor trivial if it is isomorphic with the obvious inclusion of N into matrices over N. We prove the existence of type II_1 factors M without non-trivial finite index subfactors. Equivalently, every M-M-bimodule with finite…

Let $p$ be a prime number and suppose that every maximal subgroup of a finite group is either $p$-nilpotent or has prime index. Such group need not be $p$-solvable, and we study its structure by proving that only one nonabelian simple group…

A finite group $G$ is called monomial if every irreducible character of $G$ is induced from a linear character of some subgroup of $G$. One of the main questions regarding monomial groups is whether or not a normal subgroup $N$ of a…

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…

For a pair of groups $G, H$ we study pairs of actions $G$ on $H$ and $H$ on $G$ such that these pairs are compatible and non-abelian tensor products $G \otimes H$ are defined.

We investigate the multiplication group of a special class of quasigroup called AG-group. We prove some interesting results such as: the multiplication group of an AG-group of order n is non-abelian group of order 2n and its left section is…

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group \~G=<G,x_1,x_2,...,x_n | w=1> always contains a nonabelian free subgroup. For n=1…

Let G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the elements of G at least m times with the coset a_tG_t irredundant then [G:G_t]\le 2^{k-m}…

For a finite non-abelian group $G$ let $\rat(G)$ denote the largest ratio of degrees of two nonlinear irreducible characters of $G$. We prove that the number of non-abelian composition factors of $G$ is bounded above by…

Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this paper, we present the first examples of nonabelian non-$G$-completely reducible…

This note deals with the computation of the factorization number $F_2(G)$ of a finite group $G$. By using the M\"{o}bius inversion formula, explicit expressions of $F_2(G)$ are obtained for two classes of finite abelian groups, improving…

Let $n$ be a positive integer and $G(n)$ denote the number of non-isomorphic finite groups of order $n$. It is well-known that $G(n) = 1$ if and only if $(n,\phi(n)) = 1$, where $\phi(n)$ and $(a, b)$ denote the Euler's totient function and…

Let $G=G_1 \times G_2$ be a finite group. We know that the second cohomology group $H^2(G,\mathbb C^\times)$ is isomorphic to $H^2(G_1,\mathbb C^\times) \times H^2(G_2,\mathbb C^\times) \times Hom(G_1/G_1' \otimes_\mathbb Z G_2/G_2',…