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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,… Click to show full abstract
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:
log log;
log;
power cycle;
prime length
Journal Title: Proceedings of the Edinburgh Mathematical Society
Year Published: 2021
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Journal Title: Proceedings of the Edinburgh Mathematical Society
Year Published: 2021
Link to full text (if available)
Share on Social Media: Sign Up to like & get
recommendations!