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• W4322773721 abstract "Abstract For a positive integer k , a group G is said to be totally k -closed if for each set $$Omega $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Ω</mml:mi> </mml:math> upon which G acts faithfully, G is the largest subgroup of $$textrm{Sym}(Omega )$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtext>Sym</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that leaves invariant each of the G -orbits in the induced action on $$Omega times cdots times Omega =Omega ^k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>×</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>×</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>Ω</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> </mml:math> . 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 $$k(G)=2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that $$k(G)geqslant 3$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⩾</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> for all other finite simple groups. We determine the value for the alternating groups, namely $$k(A_n)=n-1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In addition, for all simple groups G , other than alternating groups and classical groups, we show that $$k(G)leqslant 7$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⩽</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> . Finally, if G is a finite simple classical group with natural module of dimension n , we show that $$k(G)leqslant n+2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⩽</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> if $$n geqslant 14$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>14</mml:mn> </mml:mrow> </mml:math> , and $$k(G) leqslant lfloor n/3 + 12 rfloor $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⩽</mml:mo> <mml:mo>⌊</mml:mo> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>12</mml:mn> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:math> 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." @default.
• W4322773721 created "2023-03-03" @default.
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• W4322773721 date "2023-03-02" @default.
• W4322773721 modified "2023-10-14" @default.
• W4322773721 title "Total closure for permutation actions of finite nonabelian simple groups" @default.
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• W4322773721 doi "https://doi.org/10.1007/s00605-023-01822-5" @default.
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