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Most permutations power to a cycle of small prime length
Published online by Cambridge University Press: 24 May 2021
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 $1-7/\log \log n$
.
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- Research Article
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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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