# What are all possible subdirect products of two quasisimple groups?

Let $$H$$ be a proper subdirect product of two quasisimple groups $$G_1$$ and $$G_2$$. We know from the definition of quasisimple group that all the normal subgroups of $$G_i$$ live in $$Z(G_i)$$, the center of $$G_i$$. Then applying Goursat's lemma, we know that there exists $$Z_i \leq Z(G_i)$$, such that $$G_1/Z_1 \cong G_2/Z_2$$, and $$H$$ satisfies $$\require{AMScd}$$ $$\begin{CD} 1 @>>> & Z_2 @>>> & H @>>> & G_1 @>>> & 1, \end{CD}$$ $$\begin{CD} 1 @>>> & Z_1 @>>> & H @>>> & G_2 @>>> & 1, \end{CD}$$ and $$\begin{CD} 1 @>>> & Z_1 \times Z_2 @>>> & H @>>> & G_1/Z_1 \cong G_2/Z_2 @>>> & 1. \end{CD}$$ My question is, since we require $$G_1$$ and $$G_2$$ to be quasisimple groups, what can we say about $$H$$ more than Goursat's lemma? For example, is it true that $$H \cong G_1 \times Z_2$$ or $$H \cong G_2 \times Z_1$$? In the special case where there exists a surjective homomorphism $$\phi: G_1 \rightarrow G_2$$, I could show that $$H \cong G_1 \times Z_2$$.

Edit on Aug. 16: As pointed out by @DerekHolt, it is not necessarily true that $$H \cong G_1 \times Z_2$$ or $$H \cong G_2 \times Z_1$$. Now I wonder whether this weaker version is true: $$H \cong G \times Z$$, where $$G$$ is a perfect central extension of both $$G_1$$ and $$G_2$$, and $$Z$$ is a simutaneous subgroup of both $$Z_1$$ and $$Z_2$$. Or even weaker: is $$[H, H]$$ a perfect group?

• It is not clear to me what you are asking. What do you mean by "all the central extensions"? The possibilities are clearly limited by $G_1$ and $G_2$. For example, if we take $G_1=G_2=A_5$, then we cannot have $H = {\rm SL}(2,5)$. But note that $H$ need not be quasisimple. Aug 16, 2021 at 7:04
• @DerekHolt Thanks for asking for clarification! I have updated my question. Does this help? Aug 16, 2021 at 7:27
• I still find "can we say more about $H$?" too vague. Could you ask something more specific? What sort of thing might you want to say about $H$? Aug 16, 2021 at 8:02
• @DerekHolt Good question. I was wondering whether $H$ should always be a semi-direct of two groups, or even a direct product of two groups. Like in the example I mentioned, $H$ is a direct product if there is a surjective homomorphism from $G_1$ to $G_2$. Aug 16, 2021 at 8:16
• We can have $G_1 \cong G_2 \cong H$, which is not a (nontrivial) direct product. I think it is either quasisimple or a direct product of a central subgroup with a quasisimple group. Aug 16, 2021 at 8:49

The answer to your question "is $$H \cong G_1 \times Z_2$$ or $$Z_1 \times G_2$$" is no (i.e. not necessarily).
As an example, the Schur Multiplier of $$A_6$$ is cyclic of order $$6$$, so we take $$G_1$$ and $$G_2$$ to be $$2$$-fold and $$3$$-fold coverings of $$A_6$$, respectively (i.e. $$2.A_6$$ and $$3.A_6$$ in ATLAS notation).
Now if we take $$Z_1 = Z(G_1)$$ (of order $$2$$) and $$Z_2 = Z(G_2)$$ (of order $$3$$), then $$H = 6.A_6$$ is the full covering group of $$A_6$$.
• Thanks! I guess similar construction works for $G_1 = SU(6)/\mathbb{Z}_3$, $G_2 = SU(6)/\mathbb{Z}_2$, and $H = SU(6)$. Aug 16, 2021 at 23:23