Leinster Perfect Groups

Perfect number is an ancient object of study. A positive integer
Tom Leinster, University of Edinburgh studied some groups taking the analogues properties of perfect numbers. In his paper “Perfect number and groups” (arXiv:math/0104012vl[math.GR]1Apr 2001) he defined a class of groups and named them “Perfect Group”. He defined the group as : “A group G is said to be perfect if sum of the order of its normal subgroups is twice the order of the group.”
Originally Perfect Group implies some other class of group. Leinster asks his apology to name these groups as Perfect Groups. Probably the reason of naming to show the analogues properties with Perfect Numbers. From now on we will call such groups as Leinster Perfect Groups.
Let us look into some examples of Leinster Perfect Groups. We are well aware of the cyclic group $$mathbb{Z}n
The group of all isometries of a regular n-sided polygon is called Dihedral Group of order
Tom Leinster showed that when
Define
And further he gave some examples of Leinster Perfect group such as
$$Q_{12},Q_{20}times mathbb{Z}{19},Q{224}times A_5times mathbb{Z}{43}times mathbb{Z}{11}$$ etc.
Till now we have seen examples of even order Leinster Perfect groups only. In mathoverflow, Leinster asked whether there is any odd order Leinster Perfect Group and the only example (till now) was first discovered by Fransḉois Bruhault one day after the question was asked. This group is given by $$G=(mathbb {Z}{127}Delta mathbb{Z}_7)timesmathbb{Z}{3^4.11^2.19^2.113}
Further research is going on this particular class of groups and following are some important results (till date) on Perfect Groups:
- Every Nilpotent Quotient of a Leinster group is cyclic.
- The Generalized Quaternion group
is Leinster if and only if . - No finite semi-simple group is Leinster group.
- There is no Leinster group of order
where and are primes.