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) does not always 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

Statement

It is possible to have a group $G$ such that the derived subgroup $[G,G]$ is not a local powering-invariant subgroup of $G$. Specifically, it is possible that there exists an element $h\in [G,G]$ and a natural number $n$ such that there exists a unique element $u\in G$ satisfying $u^{n}=h$, and despite this, $u\notin H$.

We can choose $G$ to be a metacyclic group. We could also choose $G$ 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:

$G:=\langle a,x\mid xax^{-1}=a^{-1},x^{2}=e\rangle$

where $e$ denotes the identity of $G$. We find that:

$[G,G]=\langle a^{2}\rangle$

is an infinite cyclic group.

Now consider the element $h=a^{2}$. Let $n=2$. We note that all elements outside $\langle a\rangle$ have order two, hence any element $u$ with $u^{2}=h$ must be inside $\langle a\rangle$. The only possibility is thus $u=a$, which is outside $H$. Thus, the element $h=a^{2}$ has a unique square root in $G$, but this is not in $H$, 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.