I'm having trouble finding a textbook that discusses Goursat's Lemma on subgroups of a direct product of groups. I've looked in several standard Algebra textbooks and I've only seen it in Serge Lang's "Algebra" as an exercise.

Is it more commonly known by another name, or perhaps subsumed by a more commonly-taught theorem?

Bonus points, but not required: if not, why isn't it included in these texts? The direct product is one of the standard first constructions and it seems like one of the first questions one would ask is "what is known about the subgroups of $G \times H$?"

  • $\begingroup$ If $K\leq G\times H$, then $K$ is a subdirect product of $\pi_G(K)\times\pi_H(K)$, so Goursat's Lemma applies to $K\leq\pi_G(K)\times\pi_H(K)$. There is no separate statement. In other words, it is subsumed by itself. $\endgroup$ – Arturo Magidin May 3 at 15:01
  • $\begingroup$ By "subsume" I simply meant that perhaps Goursat's Lemma is implied by something else that is more commonly-known. I don't mean to imply that it doesn't give a sufficiently extensive resolution of the question that it answers. But I think I'll add your point to the wikipedia article, thanks. $\endgroup$ – Jonathan Rayner May 3 at 15:39
  • $\begingroup$ Yes, I understood what you meant by "subsume". But Goursat's lemma is not a special case of a more general theorem: it is the more general theorem: the structure of every subgroup of $G\times H$ is described by an application of Goursat's lemma: it is the "graph" of an isomorphism of a quotient of a subgroup of $G$ with a quotient of a subgroup of $H$. $\endgroup$ – Arturo Magidin May 3 at 15:44
  • $\begingroup$ As I understand it, either you make the weak claim that "GL sufficiently characterizes the subgroups of $G \times H$," or you are making the strong claim "No theorems exist or can exist that imply GL." In the case of the weak claim, my question is simply a social one - I accept your point, but I wonder if historically GL has been grouped under some other set of ideas and renamed, or has an equivalent formulation and that is why I don't see it in standard texts. In the case of the strong claim, I think that this requires proof and would make an incredible answer. $\endgroup$ – Jonathan Rayner May 3 at 16:04
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    $\begingroup$ Goursat's Lemma is also discussed in Marshal Hall's "Theory of Groups", Chapter 5, Section 5, page 63 of the Chelsea edition, Theorem 5.5.1, with a full proof given. But it's not labeled by name. $\endgroup$ – Arturo Magidin May 3 at 16:15

Goursat's Lemma is in several group theory textbooks, but not always by name.

It appears in Marshal Hall's classic The Theory of Groups. In the AMS Chelsea Publications edition, it is Theorem 5.5.1 on pages 63-64, and it occurs in the index under "subdirect product". It appears in W.R. Scott's classic Group Theory in Section 4.3, "Subdirect Products", as the statement labeled 4.3.1 (pp 71, Prentice Hall, 1964 printing). It shows up in Hungerford's Algebra, but in the section on Rings, discussing subdirect irreducibility, and there is usually a robust discussion of the concept in books on Universal Algebra when discussing subdirect representations; e.g., Gratzer's Universal Algebra.

Goursat's Lemma is not a special case of a more general theorem describing subgroups of a direct product: it is the general theorem that describes subgroups of a direct product. It may not look like it on first sight, but it really is.

To be explicit, here is Goursat's Lemma:

Goursat's Lemma. Let $G$ and $H$ be groups, and let $K$ be a subdirect product of $G$ and $H$; that is, $K\leq G\times H$, and $\pi_G(K)=G$, $\pi_H(K)=H$, where $\pi_G$ and $\pi_K$ are the projections onto the first and second factor, respectively from $G\times H$. Let $N_2=K\cap\mathrm{ker}(\pi_G)$ and $N_1=K\cap\mathrm{ker}(\pi_H)$. Then $N_2$ can be identified with a normal subgroup $N_G$ of $G$, $N_1$ can be identified with a normal subgroup $N_H$ of $H$, and the image of $K$ in $G/N_G\times H/N_H$ is the graph of an isomorphism $G/N_G \cong H/N_H$.

Another way to think about Goursat's Lemma is that we start with a quotient $G/N$ of $G$, and a quotient $H/M$ of $H$. If $\varphi\colon G/N\to H/M$ is an isomorphism, then $\varphi$ induces a subgroup of $G\times H$, by $$ K_{\varphi} = \{ (g,h)\in G\times H\mid \varphi(gN) = hM\}.$$ It is not hard to verify that $K_{\varphi}$ is a subdirect product of $G\times H$, and Goursat's Lemma is the statement that every subdirect product of $G\times H$ arises in this way:

Goursat's Lemma (restatement). Let $G$ and $H$ be groups, let $N\triangleleft G$, $M\triangleleft H$, and let $\varphi\colon G/N\to H/M$ be an isomorphism. Then $\varphi$ gives rise to a subgroup $$ K_{\varphi} = \{ (g,h)\in G\times H\mid \varphi(gN) = hM\}$$ with $\pi_G(K_{\varphi}) = G$ and $\pi_H(K_{\varphi}) = H$. Moreover, every subdirect product of $G\times H$ (every $K\leq G\times H$ with $\pi_G(K)=G$ and $\pi_H(K)=H$) arises in this way.

Now let $K$ be an arbitrary subgroup of $G\times H$, not necessarily a subdirect product. What can we say about $K$? Well, we can apply Goursat's Lemma, but not to $G\times H$, but rather to $\pi_G(K)\times\pi_H(K)$. That is, any subgroup of $G\times H$ is a subdirect product of a subgroup of $G\times H$ that is of the form $G_1\times H_1$, with $G_1\leq G$ and $H_1\leq H$. And so we can apply Goursat's Lemma to $K\leq G_1\times H_1$.

So Goursat's Lemma yields the following:

Goursat's Lemma for arbitrary sugroups of a direct product. Given groups $G$ and $H$, if $G_1\leq G$, $H_1\leq H$, $N\triangleleft G_1$, $M\triangleleft H_1$, and $\varphi\colon G_1/N \to H_1/M$ is an isomorphism, then $\varphi$ gives rise to a subgroup of $G\times H$, "the graph of $\varphi$", by $$ K_{\varphi} = \{ (g,h)\in G\times H\mid g\in G_1, h\in H_1, \varphi(gN)=hM\}.$$ Moreover, every subgroup of $G\times H$ arises in this way.


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