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Enumerating Transitive Finite Permutation Groups
Author(s) -
Lucchini Andrea
Publication year - 1998
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/s0024609398004846
Subject(s) - mathematics , transitive relation , combinatorics , conjecture , permutation group , permutation (music) , primitive permutation group , cyclic permutation , symmetric group , infinity , degree (music) , prime (order theory) , prime power , mathematics subject classification , discrete mathematics , group (periodic table) , mathematical analysis , chemistry , organic chemistry , physics , acoustics
Denote by f ( n ) the number of subgroups of the symmetric group Sym( n ) of degree n , and by f trans ( n ) the number of its transitive subgroups. It was conjectured by Pyber [ 9 ] that almost all subgroups of Sym( n ) are not transitive, that is, f trans ( n )/ f ( n ) tends to 0 when n tends to infinity. It is still an open question whether or not this conjecture is true. The difficulty comes from the fact that, from many points of view, transitivity is not a really strong restriction on permutation groups, and there are too many transitive groups [ 9 , Sections 3 and 4]. In this paper we solve the problem in the particular case of permutation groups of prime power degree, proving the following result. 1991 Mathematics Subject Classification 20B05, 20D60.