Leinster group

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Definition

In words

A group is called a Leinster group if the sum of the orders of its proper normal subgroups is equal to its order.

In symbols

A group $G$ is called a Leinster group if $\sum _{N\trianglelefteq G,N\neq G}|N|=|G|$.

Other names

Leinster groups are named after Tom Leinster, a mathematician who has investigated them extensively. Leinster groups are sometimes called perfect groups, due to the similar definition of perfect numbers. However, the term perfect group usually has another meaning. Leinster also called them immaculate groups.

Examples

Let $N$ be a perfect number (a number whose proper divisors sum to itself). Then the cyclic group of order $N$ is a Leinster group. See cyclic group is Leinster group if and only if of perfect order for a proof.

There is a non-abelian Leinster group of order 355433039577. This is the smallest known non-abelian Leinster group of odd order.

The smallest Leinster groups (those of order 6, 12, 28) are cyclic group:Z6, dicyclic group:Dic12, cyclic group:Z28.

External links

A086792 in the OEIS, the orders of the Leinster groups, begins 6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, 1656, 1680, 1980....