Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Intersecting families in the alternating group and direct product of symmetric groups. | Intersecting families in the alternating group and direct product of symmetric groups Cheng Yeaw Ku Department of Mathematics California Institute of Technology Pasadena CA 91125 USA cyk@. Tony W. H. Wong Department of Mathematics The Chinese University of Hong Kong Hong Kong tonywhwong@. Submitted Oct 27 2006 Accepted Mar 6 2007 Published Mar 15 2007 Mathematics Subject Classification 05D99 Abstract Let Sn denote the symmetric group on n 1 . n . A family I c Sn is intersecting if any two elements of I have at least one common entry. It is known that the only intersecting families of maximal size in Sn are the cosets of point stabilizers. We show that under mild restrictions analogous results hold for the alternating group and the direct product of symmetric groups. 1 Introduction Let Sn or Sym n denote the symmetric group on the symbol-set n 1 . n . Throughout the product or composition of two permutations g h 2 Sn denoted by gh will always mean do h first followed by g . We say that a family I c Sn of permutations is intersecting if x g x h x 0 for every g h 2 I . the Hamming distance dH g h x g x h x n 1 for every g h 2 I. In a setting of coding theory Deza and Frankl 5 studied extremal problems for permutations with given maximal or minimal Hamming distance. Among other results they proved that if I is an intersecting family in Sn then I n 1 . Recently Cameron and Ku 4 showed that equality holds if and only if I g 2 Sn g x y for some x y 2 n . I is a coset of a point stabilizer. This can also be deduced from a more general theorem of Larose and Malvenuto 8 about Kneser-type graphs. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R25 1 Theorem 5 4 8 Let n 2 and I be an intersecting family in Sn. Then II n 1 . Moreover equality holds if and only if I g 2 Sn g x y for some x y 2 n . Here we extend the study of intersecting families of Sn to that of the alternating group An and the direct product of symmetric groups Sni X- -X Snq. .
