Permutations with Orders Coprime to a Given Integer

Abstract

Let m be a positive integer and let ρ(m, n) be the proportion of permutations of the symmetric group Sym(n) whose order is coprime to m. In 2002, Pouyanne proved that ρ(n, m)n 1− φ(m) m ∼ κ m where κ m is a complicated (unbounded) function of m. We show that there exists a positive constant C(m) such that, for all n m, C(m) n m φ(m) m −1 ρ(n, m) n m φ(m) m −1where φ is Euler's totient function.

Untitled

“…The theory of the cycle type of random permutations of the symmetric group S n is very active, with many applications in combinatorics, group theory and number theory. A selection of applications includes • the distribution of orders of permutations (the least common multiple of cycle lengths) [1,7,10,13,22,23,24,25,26,27,28,38,50,57,61,62,63] and [40,Sec. 6];…”

Most permutations power to a cycle of small prime length

“…Instead one searches for a 'pre-p-cycle', a permutation which powers to a p-cycle. It is shown in [17,Lemma 10.2.3] that the proportion of elements in A n or S n that power to a p-cycle with n/2 < p n − 3 is asymptotically log 2/ log n, while an application of the main result Theorem 1 of [1] shows that considering only pre-p-cycles with p bounded, say p m, produces a proportion c(m)/n 1/m . Thus, to approach Seress's asymptotic proportion, the primes p must be allowed to grow unboundedly with n.…”

Cycle type of random permutations: A toolkit

“…• The distribution of orders of permutations (the least common multiple of cycle lengths) [1,7,10,13,22,23,24,25,26,27,28,37,47,54,58,59,60] and [39,Sec. 6]; • Invariable generation of the symmetric group [18,49,64] and other classical groups [56];…”