Derived subgroup not is local divisibility-closed in nilpotent group

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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 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 G (in fact, in our example, G is a group of nilpotency class two) such that the [derived subgroup]] G' is not a local divisibility-closed subgroup, i.e., there exists an element of G' that has n^{th} roots in G but no n^{th} roots in G'.

Related facts

Opposite facts

Proof

The simplest examples are: