Related papers: On two classes of dense 2-generator subgroups in $…

Using Goursat's lemma for groups, a simple representation and the invariant factor decompositions of the subgroups of the group Z_m x Z_n are deduced, where m and n are arbitrary positive integers. As consequences, explicit formulas for the…

In this work, we systematically derive explicit expressions for the Poincar\'e Group generators on arbitrary-rank tensors and spinor-tensors in $D=3+1$ and $D=2+1$ spacetimes, thus generalizing previous works in the literature for 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…

For a finite group $G$ denote by $\gamma(L(G))$ the genus of the subgroup graph of $G.$ We prove that $\gamma(L(G))$ tends to infinity as either the rank of $G$ or the number of prime divisors of $|G|$ tends to infinity.

We describe an algorithm for determining the algebraic subgroup of GL(n,C) that is defined as the closure of the group generated by a finite number of elements of GL(n,C). The algorithm avoids the use of Groebner bases and can be used on…

We show that the discrete Heisenberg group has unbounded dead-end depth with respect to every finite generating set. We also show that, in contrast, it has bounded retreat depth.

A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space $A\subseteq \mathcal B(H)$ and every sequence of unital completely positive…

The power graph $\Gamma_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have…

The supremum of lengths of strict chains of nested centralizers is called the $c$-dimension (centralizer dimension) of $G$. We prove two structure theorems for locally finite groups of finite $c$-dimension. We also prove that the…

We classify all finite $p$-groups $G$ for which |$Aut_{c}(G)$| attains its maximum value, where $Aut_{c}(G)$ denotes the group of all class preserving automorphisms of $G$ .

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$…

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C_1,...,C_n are finite cyclic groups then there exists a finite group G which is…

A combing is a set of normal forms for a finitely generated group. This article investigates the language-theoretic and geometric properties of combings for nilpotent and polycyclic groups. It is shown that a finitely generated class 2…

A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$\Pi$-property in $G$ if $G$ has a chief series $\Gamma_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant…

For a finite group we introduce a particular central extension, the unitary cover, having minimal exponent among those satisfying the projective lifting property. We obtain new bounds for the exponent of the Schur multiplier relating to…

We describe a bound on the degree of the generators for some adjoint rings on surfaces and threefolds.

For a group $G$ and a finite set $A$, denote by $\text{CA}(G;A)$ the monoid of all cellular automata over $A^G$ and by $\text{ICA}(G;A)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of…

In this paper we introduce the notion of a quasi-powerful $p$-group for odd primes $p$. These are the finite $p$-groups $G$ such that $G/Z(G)$ is powerful in the sense of Lubotzky and Mann. We show that this large family of groups shares…

Given an infinite linear group with a finite set of generators, we show that the shortest word length of an element of infinite order has an upper bound that depends only on the number of generators and the degree. This provides a…

The unipotent subgroup of a finite group of Lie type over a prime field Z/pZ comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present…