Leinster group

In mathematics, a Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups.
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 , 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.
Examples.
The cyclic groups whose order is a perfect number are Leinster groups.
It is possible for a non-abelian Leinster group to have odd order; an example, of order 355433039577, was constructed by François Brunault.
Other examples of non-abelian Leinster groups include certain groups of the form formula_1, where formula_2 is an alternating group and formula_3 is a cyclic group . For instance, the groups formula_4, formula_5 , formula_6 and formula_7 are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form formula_8, such as formula_9.
The possible orders of Leinster groups form the integer sequence