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.
The cyclic groups whose order is a perfect number are Leinster groups.
Other examples of non-abelian Leinster groups include certain groups of the form , where is an alternating group and is a cyclic group . For instance, the groups , , and  are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form , such as .
The possible orders of Leinster groups form the integer sequence
- 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.
- 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.
- 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.
- Leinster, Tom (2001), "Perfect numbers and groups", Eureka, 55: 17–27, arXiv:math/0104012, Bibcode:2001math......4012L
- 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).
- Weg, Yanior (2018), "Solutions of the equation (m! + 2)σ(n) = 2n⋅m! where 5 ≤ m", math.stackexchange.com. Accepted answer by Julian Aguirre.