This relationship is closest in the case of nilpotent Lie groups.

All nilpotent groups of class 3 or less are metabelian.

A nilpotent group is one that has a central series of finite length.

A central series exists if and only if the group is nilpotent.

It is also true that finite nilpotent groups are supersolvable.

A group has Fitting length 1 if and only if it is nilpotent.

In these cases elements are either nilpotent or invertible.

An ideal is nil if it consists only of nilpotent elements.

However, the singular radical of a Noetherian ring is always nilpotent.

The trace of any power of a nilpotent matrix is zero.

In a nilpotent group, every chief factor is centralized by every element.

The torsion elements in a nilpotent group form a normal subgroup.

It is a descending nilpotent series, at each step taking the minimal possible subgroup.

More generally, any triangular matrix with 0s along the main diagonal is nilpotent.

Every singular matrix can be written as a product of nilpotent matrices.

If an ideal is nilpotent, it is of course nil.

These can be particularly useful in the study of solvable groups and nilpotent groups.

This is because any such group is a nilpotent Lie group.

It also can be defined as a factor of a connected nilpotent Lie group by a lattice.

Though the examples above have a large number of zero entries, a typical nilpotent matrix does not.

In a right artinian ring, any nil ideal is nilpotent.

The roots of unity and nilpotent elements in any ring are integral over Z.

This nilpotent Lie group is also special in that it admits a compact quotient.

Conversely, every finite nilpotent group is the direct product of p-groups.

Many properties of nilpotent groups are shared by hypercentral groups.