The expected number of elements to generate a finite group with d-generated Sylow subgroups

2017/07/22 by Andrea Lucchini, Lucchini, Andrea, Mariapia Moscatiello +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #20P05 #FOS: Mathematics #Finite Group Theory Research #Genetics and Neurodevelopmental Disorders #Group Theory (math.GR) #Limits and Structures in Graph Theory #math.GR #msc:20P05

paper · pdf · doi:10.48550/arxiv.1707.07193

arxiv created 2017/07/22 · openalex publication_date 2017/07/22 · arxiv updated 2017/07/25 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

Given a finite group G, let e(G) be expected number of elements of G which have to be drawn at random, with replacement, before a set of generators is found. If all the Sylow subgroups of G can be generated by d elements, then e(G)≤ d+κ with κ∼ 2.75239495. The number κ is explicitly described in terms of the Riemann zeta function and is best possible. If G is a permutation group of degree n, then either G=S₃ and e(G)=2.9 or e(G)≤ \lfloor n/2\rfloor+κ^* with κ^* ∼ 1.606695.

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