Related papers: A Note On Transversals

For a group $G,$ let $\Gamma(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $\Gamma^*(G)$ be the subgraph of $\Gamma(G)$…

A subset $S$ of a group $G$ invariably generates $G$ if $G$ is generated by $\{ s^g(s) | s\in S\} $ for any choice of $g(s)\in G, s\in S$. In case $G$ is topological one defines similarly the notion of topological invariable generation. A…

Over each nontrivial finite group $G$, there exists a finite system of equations having no solutions in larger finite groups but having a solution in a periodic group containing $G$. We prove several similar facts about amenable, orderable,…

In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition:…

Let $F$ be a free group of arbitrary rank and let $H$ be a finitely generated subgroup of $F$. Given a pseudovariety $\mathbf{V}$ of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct…

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…

Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut(G) on the set of maximal independent subsets of G determine the indecomposable decompositions of G. G contains a direct sum of pure strongly…

For any group $G$ with subgroup $H$ and a set of representatives $T$ from the set of cosets $G/H$, we develop a rewriting system from $G$ that bequeaths a product into the set decomposition $T\times H$ of $G$, converting it into a group. In…

We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group G and a nonabelian subgroup A of G, we describe G as a…

A group is metabelian if its commutator subgroup is abelian. For finitely generated metabelian groups, classical commutative algebra, algebraic geometry and geometric group theory, especially the latter two subjects, can be brought to bear…

A generating set $S$ for a group $G$ is independent if the subgroup generated by $S\setminus \{s\}$ is properly contained in $G$, for all $s \in S.$ In this paper, we study a problem proposed by Peter Glasby: we investigate finite groups,…

Let $G$ be a finite insoluble group with soluble radical $ R(G)$. The solubility graph $\Gamma_{\rm S}(G)$ of $G$ is a simple graph whose vertices are the elements of $G\setminus R(G) $ and two distinct vertices $x$ and $y$ are adjacent if…

Given a group G, a (unital) ring A and a group homomorphism $\sigma : G \to \Aut(A)$, one can construct the skew group ring $A \rtimes_{\sigma} G$. We show that a skew group ring $A \rtimes_{\sigma} G$, of an abelian group G, is simple if…

A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We construct the JSJ decomposition of a vGBS group over abelian groups. We prove that…

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the…

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent…

We show that there is an order-preserving embedding of the additive group of rational numbers $\mathbb{Q}$ into a 2-generator group $G$. The group $G$ can be chosen to be a solvable group $G$ of length 3, which is a minimal result in the…

If G is a group with a presentation of the form < x,y|x^3=y^5=W(x,y)^2=1 >, then either G is virtually soluble or G contains a free subgroup of rank 2. This provides additional evidence in favour of a conjecture of Rosenberger.

The concept of `topological right transversal' is introduced to study right transversals in topological groups. Given any right quasigroup $S$ with a Tychonoff topology $T$, it is proved that there exists a Hausdorff topological group in…

Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects every conjugacy class of involutions of G.