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Operation in group theory

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:
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 (also known as splitting extension).

The dihedral group with 2n 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:

The Euclidean group of all rigid motions (isometries) of the plane (maps such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in ) is isomorphic to a semidirect product of the abelian group (which describes translations) and the group O(2) of orthogonal 2 × 2 matrices (which describes rotations and reflections that keep the origin fixed). 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 (i. e. applying the conjugate of the original translation). 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 O(2), and that the corresponding homomorphism is given by matrix multiplication: .

The orthogonal group O(n) of all orthogonal real n × n matrices (intuitively the set of all rotations and reflections of n dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of n dimensional space) and If we represent as the multiplicative group of matrices , where R is a reflection of n dimensional space that keeps the origin fixed (i. e., an orthogonal matrix with determinant –1 representing an involution), then is given by for all H in and N in SO(n). In the non trivial case (H is not the identity) this means that φ(H) is conjugation of operations by the reflection (in 3 dimensional space a rotation axis and the direction of rotation are replaced by their "mirror image").

The group of semilinear transformations on a vector space V over a field , often denoted ΓL(V), is isomorphic to a semidirect product of the linear group GL(V) (a normal subgroup of ΓL(V)), and the automorphism group of .

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.

Of course, no simple group can be expressed as a semi direct product (because they do not have nontrivial normal subgroups), but there are a few common counterexamples of groups containing a non trivial normal subgroup that nonetheless cannot be expressed as a semi direct 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.

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's an extensionIf the extension was split, then the group inwould be isomorphic to .

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 .

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

Suppose G is a semidirect product of the normal subgroup N and the subgroup H. If H is also normal in G, or equivalently, if there exists a homomorphism G → N that is the identity on N with kernel H, then G is the direct product of N and H.

The direct product of two groups N and H can be thought of as the semidirect product of N and H with respect to for all h in H.

Note that in a direct product, the order of the factors is not important, since N × H is isomorphic to H × N. This is not the case for semidirect products, as the two factors play different roles. Furthermore, the result of a (proper) semidirect product by means of a non trivial homomorphism is never an abelian group, even if the factor groups are abelian.

In general, there is no known characterization (i. e., a necessary and sufficient condition) 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.

Within group theory, the construction of semidirect products can be pushed much further. The Zappa–Szép 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. The semidirect product is a special case of the Grothendieck construction in category theory. Specifically, an action of on (respecting the group, or even just monoid structure) is the same thing as a functor

from the groupoid associated to H (having a single object *, whose endomorphisms are H) to the category of categories such that the unique object in is mapped to The Grothendieck construction of this functor is equivalent to , the (groupoid associated to) semidirect product.

Another generalization is for groupoids. This occurs in topology because if a group G acts on a space X 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 X/G. For full details see Chapter 11 of the book referenced below, and also some details in semidirect product in ncatlab.

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.