Groups occuring as derived subgroups.

Question

I want to solve this problem but I have no idea how to start it. If you know please hint me, thanks.

Suppose that $G$ is a group that has subgroup which is cyclic, characteristic and not in the center of $G$. Prove that there does not exist a group $K$ such that $K^{\prime}\cong G$, where $K^{\prime}$ is the derived (commutator) subgroup of $K$.

@sleevechen I think that it is the derived subgroup of $K$. However, I am not sure. — Crostul, May 29 '15 at 09:03
A more general result can be found here: http://math.stackexchange.com/questions/361582/which-groups-are-derived-subgroups — Seirios, May 30 '15 at 07:46

score 1 · Accepted Answer · edited May 30 '15 at 20:19

Suppose by contradiction that there exists a group $K$ such that $G \cong K'$. Then $K'$ has a cyclic characteristic subgroup $N \subseteq K'$ not contained in the center of $K'$.

Now, $N$ is characteristic in $K'$, and $K'$ is characteristic in $K$, so $N$ is normal in $K$. So, by this result $K'$ is contained in the centralizer $C_K(N)$, so $N$ is contained in the center of $K$. A contradiction.

Groups occuring as derived subgroups.

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