Leinster group

Short description: In mathematics, a group-theoretic analogue of the perfect numbers

For other uses, see Leinster group (disambiguation).

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) 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] Moreover Leinster showed that dihedral Leinster groups are in one-to-one correspondence with odd perfect numbers, so the existence of odd perfect numbers is equivalent to the existence of dihedral Leinster groups.

## 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 An×Cm, where An is an alternating group and Cm is a cyclic group. For instance, the groups A5×C15128, A6×C366776 [4], A7×C5919262622 and A10×C691816586092[5] are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form Sn×Cm, such as S3×C5.[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)

It is unknown whether there are infinitely many Leinster groups.

## 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. ↑ 1.0 1.1 1.2 1.3 Baishya, Sekhar Jyoti (2014), "Revisiting the Leinster groups", Comptes Rendus Mathématique 352 (1): 1–6, doi:10.1016/j.crma.2013.11.009 .
2. ↑ 2.0 2.1 2.2 De Medts, Tom; Maróti, Attila (2013), "Perfect numbers and finite groups", Rendiconti del Seminario Matematico della Università di Padova 129: 17–33, doi:10.4171/RSMUP/129-2, http://www.renyi.hu/~maroti/leinster1.pdf .
3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 Leinster, Tom (2001), "Perfect numbers and groups", Eureka 55: 17–27, Bibcode: 2001math......4012L, https://www.archim.org.uk/eureka/archive/Eureka-55.pdf
4. ↑ 4.0 4.1 4.2 4.3 Leinster, Tom (2011), "Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups?", MathOverflow, https://mathoverflow.net/q/54851 . Accepted answer by François Brunault, cited by (Baishya 2014).
5. ↑ Weg, Yanior (2018), "Solutions of the equation (m! + 2)σ(n) = 2n⋅m! where 5 ≤ m", math.stackexchange.com, https://math.stackexchange.com/q/2872997/407165(''m''!+++2)''σ''(''n'')+=+2''n''⋅''m''!+where+5+≤+''m''&rft.jtitle=math.stackexchange.com&rft.aulast=Weg&rft.aufirst=Yanior&rft.au=Weg, Yanior&rft.date=2018&rft_id=https://math.stackexchange.com/q/2872997/407165&rfr_id=info:sid/en.wikibooks.org:Leinster_group"> . Accepted answer by Julian Aguirre.

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