Leinster group

Template:Short description Template:Other uses In mathematics, a Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups.Template:R

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.Template:R He called them "perfect groups"Template:R and later "immaculate groups",Template:R but they were renamed as the Leinster groups by Template:Harvtxt because "perfect group" already had a different meaning (a group that equals its commutator subgroup).Template:R

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.Template:R 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.Template:R 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.Template:R

It is possible for a non-abelian Leinster group to have odd order; an example of order 355433039577 was constructed by François Brunault.Template:R

Other examples of non-abelian Leinster groups include certain groups of the form ${\mathrm{A}}_n\times {\mathrm{C}}_m$, where ${\mathrm{A}}_n$ is an alternating group and ${\mathrm{C}}_m$ is a cyclic group. For instance, the groups ${\mathrm{A}}_5\times {\mathrm{C}}_{15128}$, ${\mathrm{A}}_6\times {\mathrm{C}}_{366776}$ Template:R, ${\mathrm{A}}_7\times {\mathrm{C}}_{5919262622}$ and ${\mathrm{A}}_{10}\times {\mathrm{C}}_{691816586092}$Template:R are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form ${\mathrm{S}}_n\times {\mathrm{C}}_m$, such as ${\mathrm{S}}_3\times {\mathrm{C}}_5$.Template:R

The possible orders of Leinster groups form the integer sequence

6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, ... Template:OEIS

It is unknown whether there are infinitely many Leinster groups.

Properties

• There are no Leinster groups that are symmetric or alternating.Template:R
• There is no Leinster group of order p²q² where p, q are primes.Template:R
• No finite semi-simple group is Leinster.Template:R
• No p-group can be a Leinster group.Template:R
• All abelian Leinster groups are cyclic with order equal to a perfect number.Template:R