A theory of semiprimitive groups

TL;DR

This paper develops a comprehensive theory of semiprimitive groups, exploring their structure, quotient actions, and construction methods, while extending primitive group results and highlighting open problems in the field.

Contribution

It introduces a unified framework for semiprimitive groups, including their structure, quotient actions, and a construction method for all finite cases, extending primitive group theory.

Findings

Established the structure of semiprimitive groups

Extended primitive group results to semiprimitive groups

Provided a construction method for all finite semiprimitive groups

Abstract

A transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular. Interest in this class of groups is motivated by two sources: problems arising in universal algebra related to collapsing monoids and the graph-restrictive problem for permutation groups. Here we develop a theory of semiprimitive groups which encompasses their structure, their quotient actions and a method by which all finite semiprimitive groups are constructed. We also extend some results from the theory of primitive groups to semiprimitive groups, and conclude with open problems of a similar nature.