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Most permutations power to a cycle of small prime length

Part of: Theory of computing Multiplicative number theory Combinatorics

Published online by Cambridge University Press:  24 May 2021

S. P. Glasby ,
Cheryl E. Praeger  and
W. R. Unger
S. P. Glasby
Affiliation:
Centre for the Mathematics of Symmetry and Computation, University of Western Australia, 35 Stirling Highway, Perth6009, Australia (stephen.glasby@uwa.edu.au; cheryl.praeger@uwa.edu.au)
Cheryl E. Praeger
Affiliation:
Centre for the Mathematics of Symmetry and Computation, University of Western Australia, 35 Stirling Highway, Perth6009, Australia (stephen.glasby@uwa.edu.au; cheryl.praeger@uwa.edu.au)
W. R. Unger
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW2006, Australia (william.unger@sydney.edu.au)
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Abstract

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

Keywords

permutationscyclesprimelengthdensity

MSC classification

Primary: 05A05: Permutations, words, matrices 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
Secondary: 11N05: Distribution of primes
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 234 - 246
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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