Finite groups, 2-generation and the uniform domination number
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FINITE GROUPS, 2-GENERATION AND THE UNIFORM DOMINATION NUMBER
BY
Timothy C. Burness and Scott Harper School of Mathematics, University of Bristol, Bristol BS8 1UG, UK e-mail: [email protected], [email protected]
ABSTRACT
Let G be a finite 2-generated non-cyclic group. The spread of G is the largest integer k such that for any nontrivial elements x1 , . . . , xk , there exists y ∈ G such that G = xi , y for all i. The more restrictive notion of uniform spread, denoted u(G), requires y to be chosen from a fixed conjugacy class of G, and a theorem of Breuer, Guralnick and Kantor states that u(G) 2 for every non-abelian finite simple group G. For any group with u(G) 1, we define the uniform domination number γu (G) of G to be the minimal size of a subset S of conjugate elements such that for each nontrivial x ∈ G there exists y ∈ S with G = x, y (in this situation, we say that S is a uniform dominating set for G). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on γu (G) for all simple groups G. In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups G with γu (G) = 2, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for G. We also establish new results concerning the 2generation of soluble and symmetric groups, and we present several open problems.
Received October 29, 2018 and in revised form September 6, 2019
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T. C. BURNESS AND S. HARPER
Isr. J. Math.
Contents
1. Introduction . . . . . . . . . . . 2. Methods . . . . . . . . . . . . . 3. Soluble groups . . . . . . . . . 4. Symmetric groups . . . . . . . 5. Alternating groups . . . . . . . 6. Exceptional groups of Lie type 7. Two-dimensional linear groups 8. Classical groups . . . . . . . . . 9. Proofs of Theorems 9 and 10 . References . . . . . . . . . . . . . .
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1. Introduction Let G be a finite non-cyclic group that can be generated by two elements. It is natural to study the properties of generating pairs for G and such problems have attracted a great deal of attention over several decades, especially in the context of finite simple groups. Here we begin by introducing the generation invariants and associated probabilities that will be the main focus of this paper. In [10], Brenner and Wiegold define the spread of G, denoted s(G), to be the largest integer k such that for any nontrivial elements x1 , . . . , xk in G, there exists y ∈ G such that G = xi , y for all i. This lea
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