Permutations Which Make Transitive Groups Primitive

TL;DR

This paper characterizes primitive groups by identifying special generators that imply primitivity, revealing infinitely many such groups beyond symmetric and alternating groups, including Mathieu and linear groups.

Contribution

It introduces a method to identify primitive groups via primitive generators and classifies infinite families of such groups beyond classical examples.

Findings

Infinitely many primitive groups have a single primitive generator.

Certain Mathieu, projective linear, and affine groups are primitive with specific generators.

Symmetric and alternating groups are exceptions to the general pattern.

Abstract

In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and certain subgroups of some affine special linear groups.