Abstract
Every finite simple group can be generated by two elements, and Guralnick and Kantor proved that, moreover, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated. Thompson's group $V$ was the first finitely presented infinite simple group to be discovered. The Higman--Thompson groups $V_n$ and the Brin--Thompson groups $mV$ are two families of finitely presented groups that generalise $V$. In this paper, we prove that all of the groups $V_n$, $V_n'$ and $mV$ are $\frac{3}{2}$-generated. As far as the authors are aware, the only previously known examples of infinite noncyclic $\frac{3}{2}$-generated groups are the pathological Tarski monsters. We conclude with several open questions motivated by our results.
Highlights
A groupG is d-generated if it has set of size d
G is 2-generated and, every nontrivial element of G is contained in a generating pair
It is a remarkable fact that every finite simple group is 2-generated
Summary
G is d-generated if it has set of size d. In 2000, Guralnick and Kantor [19] resolved this long-standing question of Steinberg and proved the following This theorem has motivated a great deal of recent work on the generation of finite groups (see Section 3 of Burness’ recent survey article [12] for a good overview of this work). These latter groups are the infinite groups whose only proper nontrivial subgroups have order p for a fixed prime p; they are clearly simple and. The authors expect that collecting together uniformly stated results on the generating sets for Vn, Vn′ and mV in Section 3 will be useful for others interested in studying these groups, as this information is currently spread across the literature in terms of various different descriptions of and notation for these groups.
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