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The spread of a finite group. (English) Zbl 07331717
Let \(G\) be a finite group, then
(1) the spread \(s(G)\) of \(G\) is the largest integer \(k\) such that for any nontrivial elements \(x_1, \ldots, x_k \in G\), there exists \(y \in G\) with \(G = \langle x_i, y \rangle\) for all \(i\);
(2) the uniform spread \(u(G)\) of \(G\) is the largest integer \(k\) such that there is a conjugacy class \(\mathcal{C}\) of \(G\) with the property that for any nontrivial elements \(x_1, \ldots, x_k \in G\), there exists \(y \in \mathcal{C}\) with \(G = \langle x_i, y \rangle\) for all \(i\).
A group \(G\) is \(\frac{3}{2}\)-generated if every nontrivial element belongs to a generating pair, which is equivalent to the condition \(s(G) \geq 1\).
The notion of spread was first introduced in the 1970s by J. L. Brenner and J. Wiegold [Mich. Math. J. 22, 53–64 (1975; Zbl 0294.20035)], where results on the spread of soluble groups and certain families of simple groups are established. The more restrictive definition of uniform spread was formally introduced much more recently by M. Quick [Int. J. Algebra Comput. 16, No. 3, 493–503 (2006; Zbl 1103.20066)].
The main result proved in the paper under review is Theorem 1: Let \(G\) be a finite group. Then \(s(G) \geq 2\) if and only if every proper quotient of \(G\) is cyclic. As a corollary, it is shown that there is no finite group \(G\) such that \(s(G)=1\).
Theorem 3 provides a classification of finite groups \(G\) with \(u(G) \leq 1\). In particular (i) \(u(G)=0\) if and only if \(G\) has a noncyclic proper quotient, or \(G \simeq \mathsf{Sym}_6\) or \(C_p \times C_p\) for a prime \(p\) and (ii) \(u(G)=1\) if and only if \(G\) has a unique minimal normal subgroup \(N=T_{1} \times \ldots \times T_{k}\) with \(T_{i} \simeq \mathsf{Alt}_{6}\) where \(k \geq 2\), \(G/N\) is cyclic and \(N_{G}(T_{i})/C_{G}(T_{i})\simeq \mathsf{Sym}_{6}\) for all \(i\). As an immediate corollary, the authors deduce that if \(G\) is a finite group of even order all of whose proper quotients are cyclic, then every involution in \(G\) belongs to a generating pair. Also, they conclude that \(G=\mathsf{Sym}_6\) is the only finite almost simple group such that \(u(G)<2\), in particular \(u(\mathsf{Sym}_6)=0\) and \(s(\mathsf{Sym}_6)=2\).
The generating graph \(\Gamma(G)\) of a finite group \(G\) is the graph whose vertices are the elements of \(G\) and in which two vertices are connected by an edge if and only if they generate \(G\). The authors of this impressive paper establish a dichotomy for generating graphs: \(\Gamma(G)\) either has isolated vertices or is connected and has diameter at most 2.

MSC:
20F05 Generators, relations, and presentations of groups
20E32 Simple groups
20E28 Maximal subgroups
20G41 Exceptional groups
20P05 Probabilistic methods in group theory
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