Random generation with cycle type restrictions

Abstract

We study random generation in the symmetric group when cycle type restrictions are imposed. Given $\pi ,{\pi }^{\prime }\in S_n$, we prove that $\pi$ and a random conjugate of ${\pi }^{\prime }$ are likely to generate at least $A_n$ provided only that $\pi$ and ${\pi }^{\prime }$ have not too many fixed points and not too many $2$-cycles. As an application, we investigate the following question: For which positive integers $m$ should we expect two random elements of order $m$ to generate $A_n$? Among other things, we give a positive answer for any $m$ having any divisor $d$ in the range $3\le d\le o(n^{1/2})$.

25 pages, 2 figures

References in corpus (1)

• Permutations contained in transitive subgroups

Cited by in corpus (2)

• Intersections of Sylow 2-subgroups in symmetric groups
• Sylow subgroups for distinct primes and intersection of nilpotent subgroups