Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T00:08:02.187Z Has data issue: false hasContentIssue false
  • English
  • Français

Random Permutations: Some Group-Theoretic Aspects

Published online by Cambridge University Press:  12 September 2008

Peter J. Cameron  and
William M. Kantor
Peter J. Cameron
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London El 4NS, U.K.
William M. Kantor
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The study of asymptotics of random permutations was initiated by Erdős and Turáan, in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more group-theoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions).

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 3 , September 1993 , pp. 257 - 262
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Cameron, P. J. (1981) Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13 1–22.CrossRefGoogle Scholar
[2]Cameron, P. J. (1992) Almost all quasigroups have rank 2. Discrete Math. 106/107 111–115.CrossRefGoogle Scholar
[3]Cameron, P. J., Neumann, P. M. and Teague, D. N. (1982) On the degrees of primitive permutation groups. Math. Z. 180 141–149.CrossRefGoogle Scholar
[4]Dixon, J. D. (1969) The probability of generating the symmetric group. Math. Z. 110 199–205.CrossRefGoogle Scholar
[5]Donnelly, P. and Grimmett, G. (to appear) On the asymptotic distribution of large prime factors. J. London Math. Soc.Google Scholar
[6]Erdős, P. and Rényi, A. (1963) Asymmetric graphs. Acta Math. Acad. Sci. Hungar. 14 295–315.CrossRefGoogle Scholar
[7]Erdős, P. and Turán, P. (1965) On some problems of a statistical group theory, I. Z. Wahrschein lichkeitstheorie und verw. Gebeite. 4 175–186.CrossRefGoogle Scholar
[8]Erdős, P. and Turán, P. (1967) On some problems of a statistical group theory, II. Acta Math. Acad. Sci. Hungar. 18 151–163.CrossRefGoogle Scholar
[9]Erdős, P. and Turán, P. (1967) On some problems of a statistical group theory, III. Acta Math Acad. Sci. Hungar. 18 309–320.CrossRefGoogle Scholar
[10]Erdős, P. and Turán, P. (1968) On some problems of a statistical group theory, IV. Acta Math Acad. Sci. Hungar. 19 413–435.CrossRefGoogle Scholar
[11]Frucht, R. (1938) Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compositio Math. 6 239–250.Google Scholar
[12]Kantor, W. M. (1988) Algorithms for Sylow p-subgroups and solvable groups. Computers in Algebra. (Proc. Conf. Chicago 1985), Dekker, New York77–90.Google Scholar
[13]Kantor, W. M. (to appear) Automorphisms and isomorphisms of symmetric and affine designs. J. Algebraic Combinatorics.Google Scholar
[14]Liebeck, M. W. and Saxl, J. (1991) Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces. Proc. London Math. Soc. (2). 63 266–314.CrossRefGoogle Scholar
[15]Luczak, T. and Pyber, L. (to appear) Combinatorics, Probability and Computing.Google Scholar
[16]Smith, J. D. H. (1986) Representation Theory of Infinite Groups and Finite Quasigroups, Sém. Math. Sup., Presses Univ. Montréal, Montréal.Google Scholar