On Leinster groups of order pqrs

TL;DR

This paper classifies Leinster groups of order p^2qr and proves that no groups of order pqrs are Leinster, providing specific isomorphism types for the former case.

Contribution

It characterizes Leinster groups of order p^2qr and establishes the non-existence of Leinster groups of order pqrs.

Findings

Leinster groups of order p^2qr are isomorphic to Q_{20}×C_{19} or Q_{28}×C_{13}.

No Leinster groups exist of order pqrs.

Provides a complete classification for these specific group orders.

Abstract

A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups equals twice the order of the group itself. Let $p<q<r<s$ be primes. We prove that if $G$ is a Leinster group of order $p^2qr$, then $G \cong Q_{20}\times C_{19}$ or $Q_{28} \times C_{13}$. We also prove that no group of order $pqrs$ is Leinster.