# Derived subgroup not is local divisibility-closed in nilpotent group

From Groupprops

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., derived subgroup) doesnotalways satisfy a particular subgroup property (i.e., local divisibility-closed subgroup)

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Statement

It is possible to have a nilpotent group (in fact, in our example, is a group of nilpotency class two) such that the [derived subgroup]] is *not* a local divisibility-closed subgroup, i.e., there exists an element of that has roots in but no roots in .

## Related facts

### Opposite facts

- Derived subgroup is divisibility-closed in nilpotent group, hence it is also a powering-invariant subgroup and quotient-powering-invariant subgroup (using that normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication).

## Proof

The simplest examples are:

- is dihedral group:D8, so is the center of dihedral group:D8. The non-identity element of has two square roots in but none in .
- is quaternion group, so is the center of quaternion group. The non-identity element of has six square roots in but none in .
- For any prime, is the semidirect product of cyclic group of prime-square order and cyclic group of prime order. is a group of prime order, and it lives inside a cyclic subgroup of order , hence its non-identity elements have roots, none of which are in .