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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 $1-7/\log \log n$
.
