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Enumerating Transitive Finite Permutation Groups

Enumerating Transitive Finite Permutation Groups Denote by f(n) the number of subgroups of the symmetric group Sym(n) of degree n, and by ftrans(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, ftrans(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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Enumerating Transitive Finite Permutation Groups

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References (11)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S0024609398004846
Publisher site
See Article on Publisher Site

Abstract

Denote by f(n) the number of subgroups of the symmetric group Sym(n) of degree n, and by ftrans(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, ftrans(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.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1998

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