Orbits of Sylow subgroups of finite permutation groups

We say that a finite group G acting on a set Ω has Property (∗)p for a prime p if Pω is a Sylow p-subgroup of Gω for all ω ∈ Ω and Sylow p-subgroups P of G. Property (∗)p arose in the recent work of Tornier (2018) on local Sylow p-subgroups of Burger-Mozes groups, and he determined the values of p for which the alternating group An and symmetric group Sn acting on n points has Property (∗)p. In this paper, we extend this result to finite 2-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy Property (∗)p for an allowable prime p.

[1]  C. Praeger.  On the Sylow subgroups of a doubly transitive permutation group. II , 1974, Bulletin of the Australian Mathematical Society.

[2]  M. Burger,et al.  Groups acting on trees: From local to global structure , 2000 .

[3]  P. B. Kleidman.  The maximal subgroups of the Chevalley groups G2(q) with q odd , 1988 .

[4]  Cheryl E. Praeger,et al.  Abundant p-singular elements in finite classical groups , 2014 .

[5]  On the Sylow subgroups of a doubly transitive permutation group , 1974 .

[6]  D. Gorenstein,et al.  The Classification of the Finite Simple Groups , 1983 .

[7]  Martin W. Liebeck,et al.  The Subgroup Structure of the Finite Classical Groups , 1990 .

[8]  R. Carter.  THE SUBGROUP STRUCTURE OF THE FINITE CLASSICAL GROUPS (London Mathematical Society Lecture Note Series 129) , 1991 .

[9]  Prime localizations of Burger–Mozes-type groups , 2017 .

[10]  Abhinav Shrestha.  CLASSIFICATION OF FINITE SIMPLE GROUPS , 2010 .

[11]  D. Gorenstein,et al.  The Classification of the Finite Simple Groups, Number 2 , 1995 .

[12]  R. Dye.  Interrelations of symplectic and orthogonal groups in characteristic two , 1979 .

[13]  P. Cameron.  Permutation groups , 1996 .

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

[15]  Cheryl E. Praeger,et al.  Proportions of elements with given 2-part order in finite classical groups of odd characteristic , 2010, 1007.2983.

[16]  P. Müller.  Finite Permutation Groups , 2013 .

[17]  Michael Giudici,et al.  Classical Groups, Derangements and Primes , 2016 .

[18]  Cheryl E. Praeger,et al.  Permutation Groups and Cartesian Decompositions , 2018 .

[19]  Cheryl E. Praeger,et al.  On the O'Nan-Scott theorem for finite primitive permutation groups , 1988, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.