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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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- Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society
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