What underlies the fact $K,H\vartriangleleft G,K\cap H=\mathbf{0},KH=G\implies K\times H\cong G$?

Question

Let $K,H\vartriangleleft G$ be two normal subgroups. I know that if $K\cap H=\mathbf{0}$ and $KH=G$ then $K\times H\cong G$.

I'm trying to understand what is it about the category of groups which makes this statement true. Since intersection and multiplication of normal subgroups are respectively their meet and join in the subobject poset of $G$, the conditions are $K\wedge H=\bf 0$ and $K\vee H=G$.

For linear categories, we may use the fact binary unions are effective to calculate the union as the pushout of $K,H$ along $K\wedge H$, which is their coproduct=biproduct=product, and we're done.

But groups have no biproducts. So what's the story from the categorical perspective?

Added. Just after asking this question I realized I should check Borceux and Bourne's book on protomodular categories. There lies a section on complemented subobjects in unital categories. Proposition 1.12.4 gives almost everything I need - I think everything except union. Thus, I think my question reduces to asking whether subobjects are complemented if and only if they are normal (kernels of something) and are complemented in the naive sense that their meet and join are respectively zero and everything in the subobject poset.

Answer 1

What makes the statement true is that the category of groups is not only pointed with finite limits, but also protomodular, i.e.

1. split epimorphisms have pullbacks
2. pullbacks of split epimorphisms reflect isomorphisms

The relevant proposition is then Proposition 3.3.2 in Borceux and Bourn's Mal'cev, Protomodular, and Homological Categories. Reworded to match the notation of your question, it states

Proposition 3.3.2. Consider a pair of normal subobjects $H\hookrightarrow G\hookleftarrow K$ with $H\cap G=0$ in a pointed protomodular category with finite limits. Then

1. There is a unique morphism $H\times K\xrightarrow{t} G$ so that $H\hookrightarrow G\hookleftarrow K$ factor as the induced inclusions $H\hookrightarrow H\times K\hookleftarrow K$ followed by $H\times K\xrightarrow{t}G$.
2. Furthermore, $H\times K\xrightarrow{t} G$ is a normal subobject, so the union of $H$ and $K$ in the poset of subobjects of $G$.

Protomodularity is the topic of Chapter 3 of the book, it's introduced in Section 3.1., normality is discussed in Section 3.2., and the above proposition is in Section 3.3.

Note that that Theorem 3.1.6. characterizes which algebraic theories have protomodular categories of models

Theorem 3.1.6. The category of models of an algebraic theory is protomodular if and only if it can be axiomatized so that

• there are $n$ constants $e_1,\dots,e_n$
• there are $n$ binary "division'' operations for which the constants are units, i.e. $\phi_i$ with $\phi_i(x,x)=e_i$
• there is a $n+1$-ary "multiplication" operation for which the division operators give a kind of multilinear inverse in that $\theta (\phi_1(x,y),\dots,\phi_n(x,y),y)=y$.

The necessary and sufficient conditions to get a pointed protomodular category of models (to which the proposition applies) is that the constants $e_1,\dots,e_n$ are all the same constant $e$.

In particular, a simple class of algebras whose models are pointed protomdular categories have

• a unique constant constant $e$
• a binary "multiplication" operation and a binary "division" operation so that $x/x=e$ and $x\cdot(y/x)=y$.

So basically, the simple case is that your theory should admit a (one-sided) group operation without the associativity conditions (so an unsymmetric loop). The more complicated case is that an $n+1$-ary operations has $n$ divisions as above.