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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 $Gamma(G)$ that is induced by all the vertices of $Gamma(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $Gamma^*(G)$ is connected. Moreover $mathrm{diam}(Gamma^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $mathrm{diam}(Gamma^*(G))=infty$ otherwise. If $F$ is the free group of rank 2, then $Gamma^*(F)$ is connected and we deduce from $mathrm{diam}(Gamma^*(mathbb{Z}times mathbb{Z}))=infty$ that $mathrm{diam}(Gamma^*(F))=infty.$
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We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.
We study the automorphisms of a graph product of finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition of Aut* W in which one of the factors is Inn W. We also give a number of applications, some of which are geometric in nature.
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 groups. In this paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to $Z_3times Z_{3^k}$ or $Z_3times Z_3times Z_p$ where $kge 1$ and $p$ is a prime. In addition, we prove that $Z_2times Z_2times Z_p$ is a Schur group for every prime $p$.
Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $sum_{pin pi(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $pi(G)$ the set of prime numbers dividing the order of $G$.
In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
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