TITLE: Which groups are derived subgroups?
QUESTION [11 upvotes]: Let $G$ be a group. When is there a group $H$ such that $G$ is isomorphic to its derived subgroup $H'$?
I only know that there is not always such a $H$; for instance, no group has its derived subgroup isomorphic to $\mathfrak{S}_n$ for $n \geq 4$ and $n \neq 6$.

REPLY [5 votes]: Here's another reason for $\mathfrak{S}_3$:
Proposition: If $H$ is a group with a characteristic subgroup $K$ with abelian automorphism group, but $K$ is not contained in the center of $H$, then $H$ is not the derived subgroup of any group.
Proof: Since $K$ is characteristic in $H$, $K$ is normal in $G$. There is a homomorphism from $G$ to the automorphism group of $K$. Since the automorphism group of $K$ is abelian, we must have that $H$ is in the kernel of this homomorphism. However, that just means $K$ is contained in the center of $H$. $\square$
This shows that dihedral groups (of order at least 6) are not derived subgroups. Their subgroup $K$ of rotations are characteristic and cyclic, so the proposition applies.