# Direct product of two groups.

Let $N$ be a minimal normal subgroup of $G$. Also let $N$ and $\frac{G}{N}$ are non-abelian simple. Can we say that $G=N\times A$ where $A$ is a non-abelian simple subgroup of $G$?

• Presumably you mean to assume $G$ is finite, so you should add that hypothesis to the question. – Stephen Sep 9 '14 at 13:28

$G/C_{G}(N)$ is isomorphic to a subgroup of the automorphism group $B$ of $N$. The only nonabelian simple composition factor of $B$ is $N$ itself (this requires CFSG). Since the composition factors of $G$ are $N$ and $G/N$, it follows that $G/C_{G}(N) \cong N$. Since $N \cap C_{G}(N) = \{ 1 \}$, it follows that $C_{G}(N)$ is the required $A$.
• I am confused---is $G$ assumed finite? – Stephen Sep 9 '14 at 12:34
• @tomasz, Ah, you're right. Probably the group is meant to be finite. Andreas' point is that for a finite simple non-abelian group $G$, the classification of finite simple groups implies that the only nonabelian composition factor in $\mathrm{Aut}(G)$ is $G$ itself, so every extension of one finite non-abelian simple group by another is in fact a direct product. +1 – Stephen Sep 9 '14 at 13:27
• @tomasz, I prove that it splits, $C_{G}(N)$ being the other factor. – Andreas Caranti Sep 9 '14 at 14:26