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On the Diameter of Random Cayley Graphs of the Symmetric Group

  • L. Babai (a1) and G. L. Hetyei (a2)
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).

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[1] L. Babai and Á. Seress (1988) On the Diameter of the Cayley Graphs of the Symmetric Group. J. Combinatorial Theory, Ser. A 49 175179.

[2] L. Babai (1989) The Probability of Generating the Symmetric Group. J. Combinatorial Theory. Ser. A 52 148153.

[4] J. D. Bovey (1980) The probability that some power of a permutation has small degree. Bull. London Math. Soc. 12 4751.

[5] J. D. Dixon (1969) The Probability of Generating the Symmetric Group. Math.Z. 110 199205.

[6] P. Erdős and P. Turán (1965) On some problems of a Statistical Group-Theory I. Wahrscheinlichkeitstheorie u. verw. Geb. 4 175186.

[7] P. Erdős and P. Turán (1967) On some problems of a Statistical Group-Theory II. Acta Math. Acad. Sci. Hung. 18 151163.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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