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

Let p be a prime number. We give the explicit structure of 2- nilpotent multiplier for each finite 2-generator p-group of class two. Moreover, 2-capable groups in that class are characterized.

Fix a prime $p$ and an integer $m$ with $p> m \geq 2$. Define the family of finite groups \[ G_n :=SL_m (\mathbb{Z}/p^{n}\mathbb{Z}) \] for $n=1,2,... $. We will prove that there exist two positive constants $C$ and $d$ such that for any…

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper we consider subgroups at the next…

In this paper we explore the structure and properties of C-groups. We define a C-group as a group $G$ with $rk(G) < rk(Z(G))$ (where $rk(G)$ is the minimal cardinal of a generating set for a group $G$). Using GAP (a group theory program)…

Degree bounds for algebra generators of invariant rings are a topic of longstanding interest in invariant theory. We study the analogous question for field generators for the field of rational invariants of a representation of a finite…

Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form…

It is known that the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$ has a finite generating set. In this paper, we give a finite presentation of the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$.

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group…

In this paper we study the limit theory of numerical semigroups with two generators. We give a complete axiomatization in some cases.

Let $G$ be a finite group, define $I(G)=\{x\in G : x^{2}=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=|I(G)|$ and $c(G)=|C(G)|$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also…

The aim of this paper is to unify the theory of ends of finitely generated groups with that of ends of locally compact, metrizable and connected topological groups. In both theories one proves that, if the number of ends is finite, then it…

We present a sharp upper bound for the number of generators of a finite group in terms of the ratio between the order and the exponent.

We investigate the two classes of finite subgroups of SU(3) that are called type C and D in the book of Miller, Blichfeldt and Dickson. We present two theorems which fully determine the form of the generators in a suitable basis. After…

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…

We construct a 2-generator recursively presented group with infinite torsion length. We also explore the construction in the context of solvable and word-hyperbolic groups.

In this note, we prove that for every integer $d\geq 2$ which is not a prime power, there exists a finite solvable group $G$ such that $d\mid |G|$, $\pi(G)=\pi(d)$ and $G$ has no subgroup of order $d$. We also introduce the CLT-degree of a…

We compute the numbers g(n,2,2) of nilpotent groups of order n, of class at most 2 generated by at most 2 generators, by giving an explicit formula for the Dirichlet generating function \sum_{n=1}^\infty g(n,2,2)n^{-s}.

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the…

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

The aim of this work is to find some exact sequences on the $c$- nilpotent multiplier of a group $G$. We also give an upper bound for the $c$- nilpotent multiplier of finite $p$-groups and give the explicit structure of groups whose take…