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On Random Generation of the Symmetric Group

Published online by Cambridge University Press:  12 September 2008

Tomasz Łuczak  and
László Pyber
Tomasz Łuczak
Affiliation:
Mathematical Institute of the Polish Academy of Sciences, Poznań, Poland
László Pyber
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary
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Abstract

We prove that the probability i(n, k) that a random permutation of an n element set has an invariant subset of precisely k elements decreases as a power of k, for k ≤ n/2. Using this fact, we prove that the fraction of elements of Sn belong to transitive subgroups other than Sn or An tends to 0 when n → ∞, as conjectured by Cameron. Finally, we show that for every ∈ > 0 there exists a constant C such that C elements of the symmetric group Sn, chosen randomly and independently, generate invariably Sn with probability at least 1 − ∈. This confirms a conjecture of McKay.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 4 , December 1993 , pp. 505 - 512
Copyright
Copyright © Cambridge University Press 1993

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