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On the Diameter of Random Cayley Graphs of the Symmetric Group

Published online by Cambridge University Press:  12 September 2008

L. Babai and
G. L. Hetyei
L. Babai
Affiliation:
Department of Computer Science, University of Chicago, 1100 E 58th St, Chicago IL 60637–1504 and Eötvös University, Budapest, Hungary E-mail: laci@cs.uchicago.edu
G. L. Hetyei
Affiliation:
Department of Mathematics, M.I.T., Cambridge MA 02139 E-mail: hetyei@athena.mit.edu
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Abstract

Let σ, π be two permutations selected at random from the uniform distribution on the symmetric group Sn. By a result of Dixon [5], the subgroup G generated by σ, π is almost always (i.e. with probability approaching 1 as n → ∞) either Sn or the alternating group An. We prove that the diameter of the Cayley graph of G defined by {σ, π} is almost always not greater than exp ((½ + o(l)). (In n)2).

Information

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 3 , September 1992, pp. 201 - 208
Copyright
Copyright © Cambridge University Press 1992

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References

[1]Babai, L. and Seress, Á. (1988) On the Diameter of the Cayley Graphs of the Symmetric Group. J. Combinatorial Theory, Ser. A 49 175–179.CrossRefGoogle Scholar
[2]Babai, L. (1989) The Probability of Generating the Symmetric Group. J. Combinatorial Theory. Ser. A 52 148–153.CrossRefGoogle Scholar
[3]Babai, L., Hetyei, G., Kantor, W. M., Lubotsky, A. and Seress, Á. (1990) On the diameter of finite groups. In: Proc. 31st IEEE Symp. on Foundations of Computer Science, St. Louis MO.857–865.Google Scholar
[4]Bovey, J. D. (1980) The probability that some power of a permutation has small degree. Bull. London Math. Soc. 12 47–51.CrossRefGoogle Scholar
[5]Dixon, J. D. (1969) The Probability of Generating the Symmetric Group. Math.Z. 110 199–205.CrossRefGoogle Scholar
[6]Erdős, P. and Turán, P. (1965) On some problems of a Statistical Group-Theory I. Wahrscheinlichkeitstheorie u. verw. Geb. 4 175–186.CrossRefGoogle Scholar
[7]Erdős, P. and Turán, P. (1967) On some problems of a Statistical Group-Theory II. Acta Math. Acad. Sci. Hung. 18 151–163.CrossRefGoogle Scholar
[8]Goncharov, V. L. (1944) On the Field of Combinatory Analysis. Isvestija Akad. Nauk. SSSR, Ser. mat. 8 3–48 (Russian. English translation: (1962) Translations of the AMS, Ser. 2,19 1–46.)Google Scholar
[9]Hardy, G. H. and Wright, E. M. (1960) An Introduction to the Theory of Numbers. Clarendon Press, Oxford.Google Scholar