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