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Definition 1: A group $G$ is quasisimple if it is perfect (i.e., $G=[G,G]$, its derived subgroup) and $\operatorname{Inn}(G)$ is simple.

NB: I know that $\operatorname{Inn}(G)\cong G/Z(G)$.

Definition 2: A group $G$ is one-headed if it has a proper normal subgroup $N$ that contains every proper normal subgroup of $G$.

The Question:

Can the direct product of two quasisimple groups not be one-headed?

Thoughts:

My intuition is: no. This is because $$Z(G\times H)\cong Z(G)\times Z(H),$$ which, in my mind, means the (candidate) unique maximal normal subgroup of the direct product is, yet again, the centre; and somehow the perfect nature of the summands plays a role.

I'm fairly sure we could use Goursat's Lemma but that seems like nuking it. It is, prima facie, an unwieldy lemma.

Perhaps the question could be phrased using exact sequences as well.

I'm looking for a counterexample or a proof, please.

Shaun
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1 Answers1

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The product of quasisimple groups is never one-headed. Let $G$ be any quasisimple group and suppose that $G\times G$ is one headed. Let $N$ be the proper normal subgroup of $G\times G$ which contains every proper normal subgroup of $G\times G$. Then $N$ contains both $G\times 1$ and $1\times G$, and hence it contains $G\times G$, which is not possible.

Deif
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