Semidirect product

In group theory, a branch of mathematics, the semidirect product describes a particular method, with the given two groups, a new group can be constructed. This construction generalizes the concept of the direct product of groups.

Definition

Given two groups N and H, and a homomorphism of the group H in the group of automorphisms of N.

The Cartesian product of the sets and is the set of all pairs with and. This product, together with the linking

A group. This semidirect product as listed, as the homomorphism structure of this group essentially determined.

Unlike the direct product play in this definition, the two constructing factors play different roles in building the product. Through the Group operates H to N, not vice versa. During the direct exchange of the product in the name of factors, not the same, but is an isomorphic structure which causes exchange of the groups, the semi -formal product directly to an undefined structure. For similar reasons, an extension to more than two factors make little sense and in the literature not common. Pointedly, even if imprecise:

The semi- direct product is neither commutative nor associative.

The exterior and interior product

The constructed by the above definition of the product group must be precisely referred to as outer semidirect product, since the group G is constructed in such definition given, disjoint groups. Inside definitions, however, refer to an already given group G with a normal subgroup N and a subgroup H. As with the direct product result in internal and external definition where an internal definition is formally possible to isomorphic groups, provided for the external construction of the homomorphism is chosen appropriately. The formal prerequisite is that the groups N and H, one of which is assumed normal subgroup or subset of the same group and their intersection is the one group.

On the other hand, two groups N and H the external semidirect product formed, then the group A contains with an induced N normal subgroup and with an isomorphic to H subgroup and can be regarded as an internal semidirect product of these " copies " of N and H.

Since the inner semi direct product is based on the concept of normal subgroup of a group, it is explained in the article with the specific sub-groups in the lower section, the inner semi- direct product. There is also the connection to the external variant, which focuses on the present article explains in more detail.

Splitting Lemma

A group G is isomorphic to the semi- direct product of two groups N and H if there is a short exact sequence

And a homomorphism, so that the identity of H. The homomorphism in this case by

Be constructed.

Examples

Theory of finite groups

• The dihedral group, ie the symmetry group of a planar regular n -gon is isomorphic to the semi- direct product of the cyclic rotational symmetry group (which can be described by a cyclic permutation of the vertices of the polygon ) with a two-element cyclic group. The element σ operations in this business by

• For n > 1, the symmetric group is isomorphic to a semi- direct product of its normal subgroup ( the alternating group ) and a two-element cyclic group. The element acts on N by be interchanged in the permutation of the numbers j and k (). As an inner semidirect product considered: For n > 1, the symmetric group is a semidirect product of its normal subgroups with their generated by any transposition subgroup.
• The theorem of Schur - Zassenhaus is a criterion when one can write a finite group as a semidirect product.

Examples of applications in transformation groups

Important examples of semi- direct products are

• The Euclidean group E (n ) that the semi- direct product of the group of translations and the group of rotary reflections H = O ( n). The element denote a shift in the vector, the element stands for a rotation. The automorphism is then for each rotation and, where each vector. In the case n = 2 ( level) you can see in this example, even in a simple way that the semidirect product of two abelian groups need not be abelian: take a non-trivial elements and consider

• The Poincaré group, which is the semidirect product of the group of translations and the group of Lorentz transformations H = O ( 3,1). The element denote a shift with the vector. The automorphism is then given by the Lorentz transformation for each and every stroke. The Poincaré group is particularly important for the theory of special relativity, where she appears as invariance.