For a finite group G, let Gamma(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. We prove that if G is soluble, then the non-isolated vertices of Gamma(G) belong to a unique connected component

For a finite group G let Γ (G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. Many deep results on the generation of the finite simple groups G can be equivalently stated as theorems that ensure that Γ (G) is a rich graph, with several good properties. In this paper we want to consider Γ (G δ ) where G is a finite non-abelian simple group and G δ is the largest 2-generated power of G, with the aim to investigate whether the good generation properties of G still affect the behaviour of Γ (G δ ). In particular we prove that the graph obtained from Γ (G δ ) by removing the isolated vertices is 1-arc transitive and connected and we investigate the diameter of this graph. Moreover, some intriguing open questions will be introduced and their solutions will be exemplified for G = Alt(5).

Let G be a (topological) group. For 2 <= d epsilon N, denote by mu(d)(G) the largest m for which there exists an m-tuple of elements of G such that any of its d entries generate G (topologically). We obtain a lower bound for mu(d)(G) in the case when G is a prosolvable group. Our result implies in particular that if G is d-generated then the difference mu(d)(G) - d tends to infinity when the smallest prime divisor of the order of G tends to infinity. One of the aim of the paper is to draw the attention to an intriguing question in linear algebra whose solution would allow to improve our bounds and determine the precise value for mu(d)(G) in several relevant cases, for example when d = 2 and G is a prosolvable group. (c) 2012 Elsevier Inc. All rights reserved

For a d-generated group G we consider the graph Lambda(d,G) in which the vertices are the ordered generating d-tuples and in which two vertices (x, ... , xd) and (y1, ..., yd) are adjacent if and only if there exists I subset of {1, ..., d} such that vertical bar I vertical bar >= [d/2] and xi= yi for each i is an element of I. We prove that if G is a finite soluble, then Lambda(d,G) is connected. We consider also the "swap graph" Delta(d,G) in which two generating d-tuples are adjacent if they differ only by one entry, proving the following: if G is an arbitrary finite group and d >= d(G) + 1, then Delta(d,G) is connected

In this article, we investigate the relationship between the minimum number of proper subgroups of GL(n, q) whose union is the whole GL(n, q) and the maximum number of elements that pairwise generate GL(n, q). We show that the minimum number of proper subrings of M_n(q) whose union is the whole M_n(q) is exactly the maximum number of elements that pairwise generate M_n(q)