A subexponential bound on the cardinality of abelian quotients in finite transitive groups

TL;DR

This paper establishes a subexponential upper bound on the size of abelian quotients of finite transitive groups, resolving a longstanding open problem from 1989.

Contribution

It provides the first subexponential bound on abelian quotients in finite transitive groups, answering a question posed over three decades ago.

Findings

Largest abelian quotient size is at most 4^{n/√log n} for degree n

The result confirms the abelian quotient size grows subexponentially with group degree

Addresses an open problem from 1989 by Kovács and Praeger

Abstract

We show that, for every transitive group $G$ of degree $n\ge 2$, the largest abelian quotient of $G$ has cardinality at most $4^{n/\sqrt{\log_2 n}}$. This gives a positive answer to a 1989 outstanding question of L\'aszl\'o Kov\'acs and Cheryl Praeger.