Sharp upper bounds on the minimal number of elements required to generate a transitive permutation group

2021/02/19 by Tracey, Gareth
#20B05 #20D05 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2102.10070

Abstract

The purpose of this paper is to prove that if G is a transitive permutation group of degree n≥ 2, then G can be generated by \lfloor cn/√logn\rfloor elements, where c:=√(3)/2. Owing to the transitive group D₈∘ D₈ of degree 8, this upper bound is best possible. Our new result improves a 2018 paper by the author, and makes use of the recent classification of transitive groups of degree 48.

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• Generating minimally transitive permutation groups
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• On the base size and minimal degree of transitive groups
• Invariable generation of permutation and linear groups