Normalizers of Primitive Permutation Groups

TL;DR

This paper investigates bounds on the size of the quotient of a permutation group by its primitive normal subgroup, establishing sharp bounds and exceptions, with extensions to linear and Galois groups.

Contribution

It provides new bounds on the size of the factor group for primitive permutation groups and extends these results to linear and Galois groups, identifying specific exceptions.

Findings

|A/G| < n for primitive G, except for specific degrees

|Out(G)| < n unless G is in an infinite sequence of primitive groups

Results extend to linear and Galois groups with similar bounds

Abstract

Let $G$ be a transitive normal subgroup of a permutation group $A$ of finite degree $n$. The factor group $A/G$ can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that $|A/G| < n$ if $G$ is primitive unless $n = 3^{4}$, $5^4$, $3^8$, $5^8$, or $3^{16}$. This bound is sharp when $n$ is prime. In fact, when $G$ is primitive, $|\mathrm{Out}(G)| < n$ unless $G$ is a member of a given infinite sequence of primitive groups and $n$ is different from the previously listed integers. Many other results of this flavor are established not only for permutation groups but also for linear groups and Galois groups.