# Leinster group

In mathematics, a Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups.[1][2]

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.[3] He called them "perfect groups",[3] and later "immaculate groups",[4] 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).[2]

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.[2] 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.[3]

## Examples

The cyclic groups whose order is a perfect number are Leinster groups.[3]

It is possible for a non-abelian Leinster group to have odd order; an example, of order 355433039577, was constructed by François Brunault.[1][4]

Other examples of non-abelian Leinster groups include certain groups of the form ${\displaystyle A_{n}\times C_{m}}$, where ${\displaystyle A_{n}}$ is an alternating group and ${\displaystyle C_{m}}$ is a cyclic group . For instance, the groups ${\displaystyle A_{5}\times C_{15128}}$, ${\displaystyle A_{6}\times C_{366776}}$ [4], ${\displaystyle A_{7}\times C_{5919262622}}$ and ${\displaystyle A_{10}\times C_{691816586092}}$ [5] are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form ${\displaystyle S_{n}\times C_{m}}$, such as ${\displaystyle S_{3}\times C_{5}}$.[3]

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.[3]
• There is no Leinster group of order p2q2, where p, q are primes.[1]
• No finite semi-simple group is Leinster.[1]
• No p-group can be a Leinster group.[4]
• All abelian Leinster groups are cyclic with order equal to a perfect number.[3]

## References

1. ^ a b c d Baishya, Sekhar Jyoti (2014), "Revisiting the Leinster groups", Comptes Rendus Mathématique, 352 (1): 1–6, doi:10.1016/j.crma.2013.11.009, MR 3150758.
2. ^ a b c De Medts, Tom; Maróti, Attila (2013), "Perfect numbers and finite groups" (PDF), Rendiconti del Seminario Matematico della Università di Padova, 129: 17–33, doi:10.4171/RSMUP/129-2, MR 3090628.
3. Leinster, Tom (2001), "Perfect numbers and groups", Eureka, 55: 17–27, arXiv:math/0104012, Bibcode:2001math......4012L
4. ^ a b c d Leinster, Tom (2011), "Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups?", MathOverflow. Accepted answer by François Brunault, cited by Baishya (2014).
5. ^ Weg, Yanior (2018), "Solutions of the equation (m! + 2)σ(n) = 2nm! where 5 ≤ m", math.stackexchange.com. Accepted answer by Julian Aguirre.