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
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
As desired.
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?