# Pre-primitive permutation groups

## TL;DR

This paper explores the concept of pre-primitive permutation groups, analyzing their properties, relationships with other group classes, and describing invariant partitions across various group classes to understand their structure.

## Contribution

It introduces and investigates the notion of pre-primitivity, examining its independence from quasiprimitivity and its relation to primitivity, with detailed descriptions of invariant partitions in different group classes.

## Key findings

- Pre-primitivity and quasiprimitivity are independent properties.
- Pre-primitivity combined with quasiprimitivity implies primitivity.
- Descriptions of invariant partitions for various transitive groups.

## Abstract

A transitive permutation group $G$ on a finite set $\Omega$ is said to be pre-primitive if every $G$-invariant partition of $\Omega$ is the orbit partition of a subgroup of $G$. It follows that pre-primitivity and quasiprimitivity are logically independent (there are groups satisfying one but not the other) and their conjunction is equivalent to primitivity. Indeed, part of the motivation for studying pre-primitivity is to investigate the gap between primitivity and quasiprimitivity. We investigate the pre-primitivity of various classes of transitive groups including groups with regular normal subgroups, direct and wreath products, and diagonal groups. In the course of this investigation, we describe all $G$-invariant partitions for various classes of permutation groups $G$. We also look briefly at conditions similarly related to other pairs of conditions, including transitivity and quasiprimitivity, $k$-homogeneity and $k$-transitivity, and primitivity and synchronization.