Leinster group

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 (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] 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 ${\displaystyle \operatorname {A} _{n}\times \operatorname {C} _{m}}$, where ${\displaystyle \operatorname {A} _{n}}$ is an alternating group and ${\displaystyle \operatorname {C} _{m}}$ is a cyclic group. For instance, the groups ${\displaystyle \operatorname {A} _{5}\times \operatorname {C} _{15128}}$, ${\displaystyle \operatorname {A} _{6}\times \operatorname {C} _{366776}}$ ^[4] , ${\displaystyle \operatorname {A} _{7}\times \operatorname {C} _{5919262622}}$ and ${\displaystyle \operatorname {A} _{10}\times \operatorname {C} _{691816586092}}$ ^[5] are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form ${\displaystyle \operatorname {S} _{n}\times \operatorname {C} _{m}}$, such as ${\displaystyle \operatorname {S} _{3}\times \operatorname {C} _{5}}$. ^[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 )

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. 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. 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" (PDF), Eureka , 55: 17–27, arXiv: math/0104012 , Bibcode:2001math......4012L
4. 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) = 2n⋅m! where 5 ≤ m", math.stackexchange.com . Accepted answer by Julian Aguirre.