Let’s talk about semi-direct products! We’ll talk about the categorical significance of semi-direct products and some simple examples. But first, just what is a semi-direct product?
Take a group with a structure-respecting group action of on a group . In other words, we have a homomorphism . Contrast this with a group action of on which does not respect the group structure of , or in other terms a homomorphism . From we can form a group . We write when the context is clear. As a set this is , but we put an operation on it which is distinct from the usual group product. Instead, we define . This group is called the semi-direct product of and . Note that it is dependent on the homomorphism . If acts on trivially, then we get the ordinary direct product , since . But in general, the semi-direct product is ‘twisted’ in a way, and certainly not isomorphic to .
Let’s check that this is a group. To see the operation is associative, take and . Then
And the RHS of the two equations is seen to be equal since is a homomorphism and is a homomorphism. is an identity for : and . And lastly we need an inverse for . It will have to be of the form for some . We need , so that we can take and that will do. But we should check that : . So is a group.
Let’s look at a few examples. Take an abelian group . acts on by and . We need to require that be abelian, so that . Under this action, we get a group . We can describe it with generators and relations as where is the subgroup generated by the relations . When , we get .
Another interesting example: and . For and , put . , the Euclidian group.
It is not hard to see that where is the smallest normal subgroup containing the relations . Take the homomorphism . It’s surjective, and it can be seen to have kernel . This gives us a way of expressing the semi-direct product as a colimit, but it’s not the nicest categorical expression at hand. To see the nicer property, let be the category of groups under . It’s objects are homomorphisms , and its morphisms are morphisms such that . We also have the category of -groups, . Its objects are groups with morphisms , and its morphisms are -equivariant homomorphisms.
We can think of the objects in as special types of -groups. Every morphism induces a -group, which consists of with the -action . If we replace an object under with its corresponding , then we lose information about the homomorphism, and it can’t be recovered. But there is a “best approximation”, namely !
In categorical terms, we have a forgetful functor whose left adjoint sends objects to objects . A morphism under is sent to that same morphism in , and a -equivariant homomorphism is sent to the morphism where .
To see that this is actually an adjoint relationship, take a -group and set a map where . We’ll check the unit definition of an adjunction: for each object in and each morphism , there is a unique morphism such that .
Take an object and a -equivariant homomorphism . We must set where and . This gives . Taking and . Then
There is a nice characterization of which objects in are isomorphic in to for some -group . Recall that retract of a morphism is a morphism such that is the identity. A morphism is isomorphic in to for some -group if and only if it has a retract. For a -group , it is clear that the composition , where , and , is the identity. Conversely, suppose we have a morphism with retract . Let . Since is normal, the -group on where induces a -group on . The following diagram commutes, where is the inclusion map:
Note that is -equivariant by the construction of the action of on . Applying the adjoint correspondence, we get a commutative diagram in :
Examination of the rows reveals that they are in fact exact sequences. That in the above diagram is an isomorphism follows from the 5-lemma. N.B. the 5-lemma is usually applied in the context of abelian groups or -modules – or an abelian category, using the Mitchell’s embedding theorem. One can check, however, that the usual diagram chase works here, however. Thus we have an isomorphism in .
Exercise: This observation also allows for a nice internal characterization of a semi-direct product. Suppose that and are subgroups of a group such that (i) , (ii) , and (iii) is normal. Show that , and that there is a section . What about the converse?