# Leinster group

In mathematics, a Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups.

The Leinster groups are named after Tom Leinster, a mathematician at the University of Edinburgh, who wrote about them in a paper written in 1996 but not published until 2001. He called them "perfect groups", and later "immaculate groups", but they were renamed as the Leinster groups by De Medts & Maróti (2013), because "perfect group" already had a different meaning (a group that equals its commutator subgroup).

Leinster groups give a group-theoretic way of analyzing the perfect numbers and of approaching the still-unsolved problem of the existence of odd perfect numbers. For a cyclic group, the orders of the subgroups are just the divisors of the order of the group, so a cyclic group is a Leinster group if and only if its order is a perfect number. More strongly, as Leinster proved, an abelian group is a Leinster group if and only if it is a cyclic group whose order is a perfect number.

## Examples

The cyclic groups whose order is a perfect number are Leinster groups.

It is possible for a non-abelian Leinster group to have odd order; an example, of order 355433039577, was constructed by François Brunault.

Other examples of non-abelian Leinster groups include certain groups of the form $A_{n}\times C_{m}$ , where $A_{n}$ is an alternating group and $C_{m}$ is a cyclic group . For instance, the groups $A_{5}\times C_{15128}$ , $A_{6}\times C_{366776}$ , $A_{7}\times C_{5919262622}$ and $A_{10}\times C_{691816586092}$ are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form $S_{n}\times C_{m}$ , such as $S_{3}\times C_{5}$ .

The possible orders of Leinster groups form the integer sequence

6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, ... (sequence A086792 in the OEIS)

## Properties

• There are no Leinster groups that are symmetric or alternating.
• There is no Leinster group of order p2q2, where p, q are primes.
• No finite semi-simple group is Leinster.
• No p-group can be a Leinster group.
• All abelian Leinster groups are cyclic with order equal to a perfect number.