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On base sizes for symmetric groups
Author(s) -
Burness Timothy C.,
Guralnick Robert M.,
Saxl Jan
Publication year - 2011
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdq123
Subject(s) - mathematics , base (topology) , cardinality (data modeling) , combinatorics , primitive permutation group , symmetric group , set (abstract data type) , permutation group , permutation (music) , pointwise , discrete mathematics , cyclic permutation , mathematical analysis , computer science , physics , acoustics , data mining , programming language
A base of a permutation group G on a set Ω is a subset B of Ω such that the pointwise stabilizer of B in G is trivial. The base size of G , denoted by b ( G ), is the minimal cardinality of a base. Let G = S n or A n acting primitively on a set with point stabilizer H . In this note, we prove that if H acts primitively on {1, …, n }, and does not contain A n , then b ( G ) = 2 for all n ⩾ 13. Combined with a theorem of James, this completes the classification of primitive actions of alternating and symmetric groups which admit a base of size 2.