Leinster group

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 ${\displaystyle \operatorname {A} _{n}\times \operatorname {C} _{m}}$, where ${\displaystyle \operatorname {A} _{n}}$ is an alternating group and ${\displaystyle \operatorname {C} _{m}}$ is a cyclic group. For instance, the groups ${\displaystyle \operatorname {A} _{5}\times \operatorname {C} _{15128}}$, ${\displaystyle \operatorname {A} _{6}\times \operatorname {C} _{366776}}$ ^[4], ${\displaystyle \operatorname {A} _{7}\times \operatorname {C} _{5919262622}}$ and ${\displaystyle \operatorname {A} _{10}\times \operatorname {C} _{691816586092}}$^[5] are Leinster groups. The same examples can also be constructed with symmetric groups, i.e., groups of the form ${\displaystyle \operatorname {S} _{n}\times \operatorname {C} _{m}}$, such as ${\displaystyle \operatorname {S} _{3}\times \operatorname {C} _{5}}$.^[3]

The possible orders of Leinster groups form the integer sequence

6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, ... (sequence A086792 in the OEIS)

It is unknown whether there are infinitely many Leinster groups.

• 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]