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) 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 [math]\displaystyle{ \operatorname{A}_n \times \operatorname{C}_m }[/math], where [math]\displaystyle{ \operatorname{A}_n }[/math] is an alternating group and [math]\displaystyle{ \operatorname{C}_m }[/math] is a cyclic group. For instance, the groups [math]\displaystyle{ \operatorname{A}_5 \times \operatorname{C}_{15128} }[/math], [math]\displaystyle{ \operatorname{A}_6 \times \operatorname{C}_{366776} }[/math] [4], [math]\displaystyle{ \operatorname{A}_{7} \times \operatorname{C}_{5919262622} }[/math] and [math]\displaystyle{ \operatorname{A}_{10} \times \operatorname{C}_{691816586092} }[/math][5] are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form [math]\displaystyle{ \operatorname{S}_n \times \operatorname{C}_{m} }[/math], such as [math]\displaystyle{ \operatorname{S}_3 \times \operatorname{C}_{5} }[/math].[3]
The possible orders of Leinster groups form the integer sequence
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
- ↑ 1.0 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.
- ↑ 2.0 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.
- ↑ 3.0 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
- ↑ 4.0 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).
- ↑ 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. Accepted answer by Julian Aguirre.
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