# does infinite leinster groups indicates infinite perfect numbers?

The simplest definition i managed to find is: Leinster group is a finite group which her order equal the sum of its normal subgroups order, and as a perfect number is a positive integer that is equal to the sum of its proper positive divisors.

We know from Leinster theorem the following:

An abelian group G is Leinster group iff its a cyclic group which |G| is perfect number

So (obviously) if we know about the existence of infinite many perfect numbers we know there are infinite Leinster groups. My question is if it's works the other direction as well (means: does $\infty$ Leinster groups indicate $\infty$ abelian Leinster groups).?

thanks

• This is probably open. There's no easy way to construct an abelian Leinster group from a non-abelian Leinster group. Non-abelian Leinster groups also seem harder to understand than the abelian case. (Minor note: Leinster suggested calling such groups "immaculate" mathoverflow.net/questions/54851/… but his name apparently stuck). – JoshuaZ Sep 12 '19 at 18:47