Leinster Perfect Groups

Perfect number is an ancient object of study. A positive integer $$n$$ is said to be perfect if $$n$$ is equal to twice the sum of its divisors. For example, $$6=1+2+3+6=12$$. Hence, $$6$$ is a perfect number. The concept of prime number is first introduced by Greek mathematicians. Till now we have $$48$$ known perfect numbers.

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 $$mathbb{Z}n$$. A group $$mathbb{Z}n$$ is Leinster Perfect if and only if $$n$$ is a perfect number. So, $$mathbb{Z}_6, mathbb{Z}{28}, mathbb{Z}{496}$$ are all perfect groups.

The group of all isometries of a regular n-sided polygon is called Dihedral Group of order $$n$$ and mathematically it is defined as

$$D_{2n}=<a,b|a^n=e,b^2=e,aba^{-1}=b^{-1}>$$

Tom Leinster showed that when $$n$$ is an odd perfect number then $$D_{2n}$$ is a Leinster Perfect Group. Interestingly, till now we don’t have any example of odd perfect number. So, if you can give me an odd perfect number then I can give you the corresponding Leinster Perfect Dihedral Group.

Define $$D(G)=sum_N|N|$$, $$N$$ is a normal subgroup of $$G$$. We can show that $$D(G_1times G_2)=D(G_1)times D(G_2)$$ i.e. the function $$D$$ is multiplicative. Using the multiplicatively property of $$D$$, we can show that $$S_3times mathbb{Z}5,A_5times mathbb{Z}{15128},A_6times mathbb{Z}_{366776}$$ are also Leinster Perfect Groups. In his paper On Arithmetic Functions of Finite Groups, Prof. Ashish Kumar Das, NEHU defined the Generalized Quaternion group as:

$$Q_{4m}=<a,b|a^{2m}=1,b^2=a^m,bab^{-1}=a^{-1}>.$$

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 $$G=(mathbb {Z}{127}Delta mathbb{Z}_7)timesmathbb{Z}{3^4.11^2.19^2.113}$$ where $$Delta$$ denote the semi direct product.

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 $$Q_{4m}$$ is Leinster if and only if $$m=3$$.
• No finite semi-simple group is Leinster group.
• There is no Leinster group of order $$p^2q^2$$ where $$p$$ and $$q$$ are primes.