TY - JOUR

T1 - Total closure for permutation actions of finite nonabelian simple groups

AU - Freedman, Saul D.

AU - Giudici, Michael

AU - Praeger, Cheryl E.

N1 - Funding Information:
The research for this paper began at the 2022 Research Retreat of the Centre for the Mathematics of Symmetry and Computation of the University of Western Australia. The authors acknowledge discussions with their colleagues Vishnuram Arumugam and Alice Devillers. The first author was supported by a St Leonard’s International Doctoral Fees Scholarship and a School of Mathematics & Statistics Ph.D. Funding Scholarship at the University of St Andrews. The research of the second and third authors was supported by ARC Discovery Project Grants DP190101024 and DP190100450, respectively.
Funding Information:
The research for this paper began at the 2022 Research Retreat of the Centre for the Mathematics of Symmetry and Computation of the University of Western Australia. The authors acknowledge discussions with their colleagues Vishnuram Arumugam and Alice Devillers. The first author was supported by a St Leonard’s International Doctoral Fees Scholarship and a School of Mathematics & Statistics Ph.D. Funding Scholarship at the University of St Andrews. The research of the second and third authors was supported by ARC Discovery Project Grants DP190101024 and DP190100450, respectively.
Publisher Copyright:
© 2023, The Author(s).

PY - 2024/2

Y1 - 2024/2

N2 - For a positive integer k, a group G is said to be totally k-closed if for each set Ω upon which G acts faithfully, G is the largest subgroup of Sym (Ω) that leaves invariant each of the G-orbits in the induced action on Ω × ⋯ × Ω = Ω k. Each finite group G is totally |G|-closed, and k(G) denotes the least integer k such that G is totally k-closed. 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(An) = 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.

AB - For a positive integer k, a group G is said to be totally k-closed if for each set Ω upon which G acts faithfully, G is the largest subgroup of Sym (Ω) that leaves invariant each of the G-orbits in the induced action on Ω × ⋯ × Ω = Ω k. Each finite group G is totally |G|-closed, and k(G) denotes the least integer k such that G is totally k-closed. 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(An) = 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.

KW - Base size

KW - k-closed permutation groups

KW - Primitive groups

KW - Simple groups

UR - http://www.scopus.com/inward/record.url?scp=85149242709&partnerID=8YFLogxK

U2 - 10.1007/s00605-023-01822-5

DO - 10.1007/s00605-023-01822-5

M3 - Article

AN - SCOPUS:85149242709

SN - 0026-9255

VL - 203

SP - 323

EP - 340

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 2

ER -