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.
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 , where is an alternating group and is a cyclic group. For instance, the groups , [4], and [5] are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form , such as .[3]
The possible orders of Leinster groups form the integer sequence
It is unknown whether there are infinitely many Leinster groups.