An Erdős-Ko-Rado theorem for finite 2-transitive groups

We prove an analogue of the classical Erd?s-Ko-Rado theorem for intersecting sets of permutations in finite 2-transitive groups. Given a finite group G acting faithfully and 2-transitively on the set ? , we show that an intersecting set of maximal size in G has cardinality | G | / | ? | . This generalises and gives a unifying proof of some similar recent results in the literature.

[1]  Keivan Mallahi-Karai,et al.  On the Erdős–Ko–Rado property for finite groups , 2013 .

[2]  Nigel Boston,et al.  The proportion of fixed-point-free elements of a transitive permutation group , 1993 .

[3]  R. Guralnick,et al.  Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston–Shalev conjecture , 2015, 1508.00039.

[4]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[5]  Jun Wang,et al.  An Erdös-Ko-Rado-type theorem in Coxeter groups , 2008, Eur. J. Comb..

[6]  A. Weil.  Sur certains groupes d'opérateurs unitaires , 1964 .

[7]  László Babai,et al.  Spectra of Cayley graphs , 1979, J. Comb. Theory B.

[8]  Bahman Ahmadi,et al.  The Erdős-Ko-Rado property for some permutation groups , 2015, Australas. J Comb..

[9]  Cheng Yeaw Ku,et al.  Intersecting families of permutations , 2003, Eur. J. Comb..

[10]  Pablo Spiga,et al.  On a relation between the rank and the proportion of derangements in finite transitive permutation groups , 2015, J. Comb. Theory, Ser. A.

[11]  Roger W. Carter,et al.  Finite groups of Lie type: Conjugacy classes and complex characters , 1985 .

[12]  P. Rardin.  Weil Representations Associated to Finite Fields , 1977 .

[13]  R. Guralnick,et al.  Low‐Dimensional Representations of Special Linear Groups in Cross Characteristics , 1999 .

[14]  Chris D. Godsil,et al.  A new proof of the Erdös-Ko-Rado theorem for intersecting families of permutations , 2007, Eur. J. Comb..

[15]  I. G. MacDonald,et al.  Lectures on Lie Groups and Lie Algebras: Simple groups of Lie type , 1995 .

[16]  R. James Milgram,et al.  Finite Groups of Lie Type , 1994 .

[17]  Bahman Ahmadi,et al.  A new proof for the Erdős-Ko-Rado theorem for the alternating group , 2014, Discret. Math..

[18]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[19]  P. Diaconis,et al.  Generating a random permutation with random transpositions , 1981 .

[20]  K. Zsigmondy,et al.  Zur Theorie der Potenzreste , 1892 .

[21]  P. Cameron.  FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS , 1981 .

[22]  Karen Meagher,et al.  The Erdős-Ko-Rado Property for Some 2-Transitive Groups , 2013, 1308.0621.

[23]  H. Weyl.  Permutation Groups , 2022 .

[24]  Claudia Malvenuto,et al.  Stable sets of maximal size in Kneser-type graphs , 2004, Eur. J. Comb..

[25]  R. Guralnick,et al.  Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements , 2009, 0902.2238.

[26]  David Ellis,et al.  Setwise intersecting families of permutations , 2011, J. Comb. Theory, Ser. A.

[27]  Ernest E. Shult,et al.  On a class of doubly transitive groups , 1972 .

[28]  J. Sutherland Frame,et al.  The Character Tables for SL(3, q), SU(3, q 2), PSL(3, q), PSU(3, q 2) , 1973, Canadian Journal of Mathematics.

[29]  László Babai,et al.  Permutation groups in NC , 1987, STOC '87.

[30]  Jason Fulman,et al.  Derangements in simple and primitive groups , 2002 .

[31]  Pablo Spiga,et al.  An Erdös-Ko-Rado Theorem for the Derangement Graph of PGL3(q) Acting on the Projective Plane , 2014, SIAM J. Discret. Math..

[32]  S. Guest,et al.  A REMARK ON THE PERMUTATION REPRESENTATIONS AFFORDED BY THE EMBEDDINGS OF IN , 2013, 1304.0852.

[33]  Cheng Yeaw Ku,et al.  Intersecting Families in the Alternating Group and Direct Product of Symmetric Groups , 2007, Electron. J. Comb..

[34]  P. Tiep,et al.  Minimal characters of the finite classical groups , 1996 .

[35]  R. Guralnick,et al.  Cross characteristic representations of even characteristic symplectic groups , 2004 .

[36]  P. Erdös,et al.  INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1961 .

[37]  R. Guralnick,et al.  Derangements in subspace actions of finite classical groups , 2013, 1303.5480.

[38]  Chris Godsil,et al.  Erdős-Ko-Rado Theorems: Algebraic Approaches , 2015 .

[39]  Harold N. Ward,et al.  On Ree’s series of simple groups , 1966 .

[40]  Pablo Spiga,et al.  An Erdös-Ko-Rado Theorem for the Derangement Graph of PGL3(q) Acting on the Projective Plane , 2009, SIAM J. Discret. Math..