# Finite group generated by abelian normal subgroups may have arbitrarily large nilpotency class

## Statement

It is possible to have a finite group (more specifically, a group of prime power order for any chosen prime number ) that has arbitrarily large nilpotency class but is a group generated by abelian normal subgroups.

In fact, the examples that we choose also have the property that *every* group of prime power order can be embedded into one of these groups, hence we get the stronger statement that every finite nilpotent group is isomorphic to a subgroup of a finite group generated by abelian normal subgroups.

## Proof

`Further information: upper-triangular unipotent matrix group`

The idea is to take the group of upper triangular matrices with s on the diagonal and entries in the prime field . This group has nilpotency class . Moreover, it is generated by abelian normal subgroups parameterized by top right rectangles: for every top right rectangle, define the corresponding subgroup as the subgroup where the diagonal has 1s, entries in the rectangle are arbitrary, and all other entries are . This subgroup is elementary abelian of order , and these subgroups together generate the whole group.