Projects per year
Abstract
A generating set for a finite group G is minimal if no proper subset generates G, and m(G) denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist a,b > 0 such that any finite group G satisfies m(G)⩽a⋅δ(G)^{b}, for δ(G)=∑p primem(Gp), where Gp is a Sylow psubgroup of G. To do this, we first bound m(G) for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank 1 or 2). In particular, we prove that there exist a,b > 0 such that any finite simple group G of Lie type of rank r over the field Fpf satisfies r+ω(f)⩽m(G)⩽a(r+ω(f))^{b}, where ω(f) denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist a,b > 0 such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over F_{pf} has size at most ar^{b}+ω(f).
Original language  English 

Article number  e70 
Number of pages  10 
Journal  Forum of Mathematics, Sigma 
Volume  11 
Early online date  10 Aug 2023 
DOIs  
Publication status  Published  10 Aug 2023 
Fingerprint
Dive into the research topics of 'The maximal size of a minimal generating set'. Together they form a unique fingerprint.Projects
 1 Active

The Group Generation: From Finite to Inf: Group Generation: From Finite to Infinite
1/10/22 → 30/09/25
Project: Fellowship