The probability of generating the symmetric group with a commutator condition

TL;DR

This paper derives formulas for the likelihood that pairs of permutations with a 3-cycle commutator generate the symmetric or alternating group, revealing that this probability diminishes as the degree increases, contrasting with classical results.

Contribution

It provides explicit formulas for the counts of such permutation pairs and establishes the asymptotic behavior of their generation probability, connecting algebra, geometry, and number theory.

Findings

Probability tends to zero as n increases

Formulas involve divisor functions and Jordan's totient function

Contrasts with Dixon's classical result

Abstract

Let B(n) be the set of pairs of permutations from the symmetric group of degree n with a 3-cycle commutator, and let A(n) be the set of those pairs which generate the symmetric or the alternating group of degree n. We find effective formulas for calculating the cardinalities of both sets. More precisely, we show that #B(n)/n! is a discrete convolution of the partition function and a linear combination of divisor functions, while #A(n)/n! is the product of a polynomial and Jordan's totient function. In particular, it follows that the probability that a pair of random permutations with a 3-cycle commutator generates the symmetric or the alternating group of degree n tends to zero as n tends to infinity, which makes a contrast with Dixon's classical result. Key elements of our proofs are Jordan's theorem from the 19th century, a formula by Ramanujan from the 20th century and a technique of square-tiled surfaces developed by French mathematicians Lelievre and Royer in the beginning of the 21st century. This paper uses and highlights elegant connections between algebra, geometry, and number theory.