Semidirect product


In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol. There are two closely related concepts of semidirect product:
  • an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup.
  • an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation.
As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products.
For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product.

Inner semidirect product definitions

Given a group with identity element, a subgroup, and a normal subgroup, the following statements are equivalent:
If any of these statements holds, we say is the semidirect product of and, written
or that splits over ; one also says that is a semidirect product of acting on, or even a semidirect product of and. To avoid ambiguity, it is advisable to specify which is the normal subgroup.
If, then there is a group homomorphism given by, and for, we have.

Inner and outer semidirect products

Inner semidirect product

Let us first consider the inner semidirect product. In this case, for a group, consider a normal subgroup and another subgroup . Assume that the
conditions on the list above hold. Let denote the group of all automorphisms of, which is a group under composition. Construct a group homomorphism defined by conjugation,
In this way we can construct a group with group operation defined as
The subgroups and determine up to isomorphism, as we will show later. In this way, we can construct the group from its subgroups. This kind of construction is called an inner semidirect product.

Outer semidirect product

Let us now consider the outer semidirect product. Given any two groups and and a group homomorphism, we can construct a new group, called the outer semidirect product of and with respect to, defined as follows:
This defines a group in which the identity element is and the inverse of the element is. Pairs form a normal subgroup isomorphic to, while pairs form a subgroup isomorphic to. The full group is a semidirect product of those two subgroups in the sense given earlier.
Conversely, suppose that we are given a group with a normal subgroup and a subgroup, such that every element of may be written uniquely in the form where lies in and lies in. Let be the homomorphism given by
for all.
Then is isomorphic to the semidirect product. The isomorphism is well defined
by due to the uniqueness of the decomposition.
In, we have
Thus, for and we obtain
which proves that is a homomorphism. Since is obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule in.
The direct product is a special case of the semidirect product. To see this, let be the trivial homomorphism then is the direct product.
A version of the splitting lemma for groups states that a group is isomorphic to a semidirect product of the two groups and if and only if there exists a short exact sequence
and a group homomorphism such that, the identity map on. In this case, is given by, where

Examples

Dihedral group

The dihedral group with elements is isomorphic to a semidirect product of the cyclic groups and. Here, the non-identity element of acts on by inverting elements; this is an automorphism since is abelian. The presentation for this group is:

Cyclic groups

More generally, a semidirect product of any two cyclic groups with generator and with generator is given by one extra relation,, with and coprime, and ; that is, the presentation:
If and are coprime, is a generator of and, hence the presentation:
gives a group isomorphic to the previous one.

Symmetric group

The symmetric group where is a semidirect product of the alternating group and any 2-element subgroup generated by a single transposition.

Holomorph of a group

One canonical example of a group expressed as a semidirect product is the holomorph of a group. This is defined as
where is the automorphism group of a group and the structure map comes from the right action of on. In terms of multiplying elements, this gives the group structure

Fundamental group of the Klein bottle

The fundamental group of the Klein bottle can be presented in the form
and is therefore a semidirect product of the group of integers with addition,, with. The corresponding homomorphism is given by.

Upper triangular matrices

The group of upper triangular matrices with non-zero determinant in an arbitrary field, that is with non-zero entries on the diagonal, has a decomposition into the semidirect product
where is the subgroup of matrices with only s on the diagonal, which is called the upper unitriangular matrix group, and is the subgroup of diagonal matrices.

The group action of on is induced by matrix multiplication. If we set
and
then their matrix product is
This gives the induced group action
A matrix in can be represented by matrices in and. Hence.

Group of isometries on the plane

The Euclidean group of all rigid motions of the plane is isomorphic to a semidirect product of the abelian group and the group of orthogonal matrices. Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector. This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and, and that the corresponding homomorphism is given by matrix multiplication:.

Orthogonal group O(''n'')

The orthogonal group of all orthogonal real matrices is isomorphic to a semidirect product of the group and. If we represent as the multiplicative group of matrices, where is a reflection of -dimensional space that keeps the origin fixed, then is given by for all H in and in. In the non-trivial case this means that is conjugation of operations by the reflection.

Semi-linear transformations

The group of semilinear transformations on a vector space over a field, often denoted, is isomorphic to a semidirect product of the linear group , and the automorphism group of.

Non-examples

Of course, no simple group can be expressed as a semidirect product, but there are a few common counterexamples of groups containing a non-trivial normal subgroup that nonetheless cannot be expressed as a semidirect product. Note that although not every group can be expressed as a split extension of by, it turns out that such a group can be embedded into the wreath product by the universal embedding theorem.

Z4

The cyclic group is not a simple group since it has a subgroup of order 2, namely is a subgroup and their quotient is, so there is an extension
If instead this extension is split, then the group in
would be isomorphic to.

Q8

The group of the eight quaternions where and, is another example of a group which has non-trivial normal subgroups yet is still not split. For example, the subgroup generated by is isomorphic to and is normal. It also has a subgroup of order generated by. This would mean would have to be a split extension in the following hypothetical exact sequence of groups:
,
but such an exact sequence does not exist. This can be shown by computing the first group cohomology group of with coefficients in, so and noting the two groups in these extensions are and the dihedral group. But, as neither of these groups is isomorphic with, the quaternion group is not split. This non-existence of isomorphisms can be checked by noting the trivial extension is abelian while is non-abelian, and noting the only normal subgroups are and, but has three subgroups isomorphic to.

Properties

If is the semidirect product of the normal subgroup and the subgroup, and both and are finite, then the order of equals the product of the orders of and. This follows from the fact that is of the same order as the outer semidirect product of and, whose underlying set is the Cartesian product.

Relation to direct products

Suppose is a semidirect product of the normal subgroup and the subgroup. If is also normal in, or equivalently, if there exists a homomorphism that is the identity on with kernel, then is the direct product of and.
The direct product of two groups and can be thought of as the semidirect product of and with respect to for all in.
Note that in a direct product, the order of the factors is not important, since is isomorphic to. This is not the case for semidirect products, as the two factors play different roles.
Furthermore, the result of a semidirect product by means of a non-trivial homomorphism is never an abelian group, even if the factor groups are abelian.

Non-uniqueness of semidirect products (and further examples)

As opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if and are two groups that both contain isomorphic copies of as a normal subgroup and as a subgroup, and both are a semidirect product of and, then it does not follow that and are isomorphic because the semidirect product also depends on the choice of an action of on.
For example, there are four non-isomorphic groups of order 16 that are semidirect products of and ; in this case, is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups:
If a given group is a semidirect product, then there is no guarantee that this decomposition is unique. For example, there is a group of order 24 that can be expressed as semidirect product in the following ways:.