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

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 $G'$.

## Proof

The simplest examples are:

• $G$ is dihedral group:D8, so $G'$ is the center of dihedral group:D8. The non-identity element of $G'$ has two square roots in $G$ but none in $G'$.
• $G$ is quaternion group, so $G'$ is the center of quaternion group. The non-identity element $-1$ of $G'$ has six square roots in $G$ but none in $G'$.
• For any prime, $G$ is the semidirect product of cyclic group of prime-square order and cyclic group of prime order. $G'$ is a group of prime order, and it lives inside a cyclic subgroup of order $p^2$, hence its non-identity elements have $p^{th}$ roots, none of which are in $G$.