# Derived subgroup not is local powering-invariant

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 powering-invariant subgroup)

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

## Contents

## Statement

It is possible to have a group such that the derived subgroup is *not* a local powering-invariant subgroup of . Specifically, it is possible that there exists an element and a natural number such that there exists a unique element satisfying , and despite this, .

We can choose to be a metacyclic group. We could also choose to be a finitely generated nilpotent group, and in fact an example of a finitely generated group of nilpotency class two.

## Related facts

- Characteristic not implies powering-invariant
- Center is local powering-invariant
- Derived subgroup is divisibility-invariant in nilpotent group

## Proof

### Example of the infinite dihedral group (metacyclic example)

`Further information: infinite dihedral group`

Consider the infinite dihedral group, given by the presentation:

where denotes the identity of . We find that:

is an infinite cyclic group.

Now consider the element . Let . We note that all elements outside have order two, hence any element with must be inside . The only possibility is thus , which is outside . Thus, the element has a unique square root in , but this is not in , completing the proof.

### Example of a central product (finitely generated group of nilpotency class two)

`Further information: central product of UT(3,Z) and Z identifying center with 2Z`

In this example, the generator of the derived subgroup has a unique square root, but this lies outside the derived subgroup (though still in the center). This gives an example where the whole group is a group of nilpotency class two.