Projects per year
Abstract
For a positive integer k, a group G is said to be totally kclosed if for each set Ω upon which G acts faithfully, G is the largest subgroup of Sym (Ω) that leaves invariant each of the Gorbits in the induced action on Ω × ⋯ × Ω = Ω ^{k}. Each finite group G is totally Gclosed, and k(G) denotes the least integer k such that G is totally kclosed. We address the question of determining the closure number k(G) for finite simple groups G. Prior to our work it was known that k(G) = 2 for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that k(G) ⩾ 3 for all other finite simple groups. We determine the value for the alternating groups, namely k(A_{n}) = n 1. In addition, for all simple groups G, other than alternating groups and classical groups, we show that k(G) ⩽ 7. Finally, if G is a finite simple classical group with natural module of dimension n, we show that k(G) ⩽ n+ 2 if n⩾ 14 , and k(G) ⩽ ⌊ n/ 3 + 12 ⌋ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on n and the type of G) on the base sizes of the primitive actions of G, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.
Original language  English 

Pages (fromto)  323340 
Number of pages  18 
Journal  Monatshefte fur Mathematik 
Volume  203 
Issue number  2 
Early online date  2 Mar 2023 
DOIs  
Publication status  Published  Feb 2024 
Projects
 2 Finished

Complexity of group algorithms and statistical fingerprints of groups
Praeger, C. & Niemeyer, A.
21/02/19 → 31/12/22
Project: Research
