Dixon's asymptotic without CFSG

TL;DR

This paper proves a bound on the probability that two random elements generate a primitive subgroup smaller than the alternating group without relying on the classification of finite simple groups, and derives Dixon's asymptotic expansion.

Contribution

It provides a new proof of Dixon's asymptotic expansion without using CFSG, establishing a probability bound for subgroup generation in symmetric groups.

Findings

Probability that two random elements generate a subgroup smaller than A_n is exponentially small.

Dixon's asymptotic expansion for subgroup generation probability is confirmed without CFSG.

New bounds improve understanding of random subgroup generation in symmetric groups.

Abstract

Without using the classification of finite simple groups, we show that the probability that two random elements of $S_n$ generate a primitive group smaller than $A_n$ is at most $\exp(-c(n \log n)^{1/2})$. As a corollary we get Dixon's asymptotic expansion \[ 1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - \cdots \] for the probability that two random elements of $S_n$ (or $A_n$) generate a subgroup containing $A_n$.