Semidirect product


In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
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.

Inner and outer semidirect products

To introduce the notion of outer semidirect products, let us first consider the inner semidirect product. In this case, for a group, consider its normal subgroup and the subgroup . Let denote the group of all automorphisms of, which is a group under composition. Construct a group homomorphism defined by conjugation for all in and in. The expression is often written as for brevity. In this way we can construct a group with group operation defined as for in and in. Together,, 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.
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; that is, the presentation:
If and are coprime, is a generator of and, hence the presentation:
gives a group isomorphic to the previous one.

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. The corresponding homomorphism is given by.

Upper triangular matrices

The group of upper-triangular matrices
with non-zero determinant has a decomposition into the semi-direct product
where
and is the subgroup where the only non-zero entries are on the diagonal. This is called the group of diagonal matrices. The group action of on is induced by matrix multiplication. If we set
and
then their matrix multiplication is
This gives the induced group action is given by
A matrix in can be represented by a 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.

Crystallographic groups

In crystallography, the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group is symmorphic. Non-symmorphic space groups have point groups that are not even contained as subset of the space group, which is responsible for much of the complication in their analysis.

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:.

Existence

In general, there is no known characterization for the existence of semidirect products in groups. However, some sufficient conditions are known, which guarantee existence in certain cases. For finite groups, the Schur–Zassenhaus theorem guarantees existence of a semidirect product when the order of the normal subgroup is coprime to the order of the quotient group.
For example, the Schur–Zassenhaus theorem implies the existence of a semi-direct product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance.

Generalizations

Within group theory, the construction of semidirect products can be pushed much further. The Zappa–Szep product of groups is a generalization that, in its internal version, does not assume that either subgroup is normal.
There is also a construction in ring theory, the crossed product of rings. This is constructed in the natural way from the group ring for a semidirect product of groups. The ring-theoretic approach can be further generalized to the semidirect sum of Lie algebras.
For geometry, there is also a crossed product for group actions on a topological space; unfortunately, it is in general non-commutative even if the group is abelian. In this context, the semidirect product is the space of orbits of the group action. The latter approach has been championed by Alain Connes as a substitute for approaches by conventional topological techniques; c.f. noncommutative geometry.
There are also far-reaching generalisations in category theory. They show how to construct fibred categories from indexed categories. This is an abstract form of the outer semidirect product construction.

Groupoids

Another generalization is for groupoids. This occurs in topology because if a group acts on a space it also acts on the fundamental groupoid of the space. The semidirect product is then relevant to finding the fundamental groupoid of the orbit space. For full details see Chapter 11 of the book referenced below, and also some details in semidirect product in ncatlab.

Abelian categories

Non-trivial semidirect products do not arise in abelian categories, such as the category of modules. In this case, the splitting lemma shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.

Notation

Usually the semidirect product of a group acting on a group is denoted by or. However, some sources may use this symbol with the opposite meaning. In case the action should be made explicit, one also writes. One way of thinking about the symbol is as a combination of the symbol for normal subgroup and the symbol for the product. Barry Simon, in his book on group representation theory, employs the unusual notation for the semidirect product.
Unicode lists four variants:
Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice.
In LaTeX, the commands \rtimes and \ltimes produce the corresponding characters.