Most permutations power to a cycle of small prime length

S. P. Glasby; Cheryl E. Praeger; W. R. Unger

doi:10.1017/S0013091521000110

Most permutations power to a cycle of small prime length

S. P. Glasby, Cheryl E. Praeger, W. R. Unger

Mathematics and Statistics

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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.

Original language	English
Pages (from-to)	234-246
Number of pages	13
Journal	Proceedings of the Edinburgh Mathematical Society
Volume	64
Issue number	2
Early online date	24 May 2021
DOIs	https://doi.org/10.1017/S0013091521000110
Publication status	Published - May 2021

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10.1017/S0013091521000110

Glasby, et al., 2021 Most permutations power to a cycleAccepted author manuscript, 313 KBLicence: CC BY-NC-ND

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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.",

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AU - Unger, W. R.

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KW - Cycles

KW - Density

KW - Length

KW - Permutations

KW - Prime

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Most permutations power to a cycle of small prime length

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Complexity of group algorithms and statistical fingerprints of groups

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Most permutations power to a cycle of small prime length

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Complexity of group algorithms and statistical fingerprints of groups

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