Most permutations power to a cycle of small prime length

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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.

Original languageEnglish
Pages (from-to)234-246
Number of pages13
JournalProceedings of the Edinburgh Mathematical Society
Issue number2
Early online date24 May 2021
Publication statusPublished - May 2021

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