Semidirect Product Some Equivalent Definitions Elementary Facts

Let G be a group, N a normal subgroup of G and H a subgroup of G. The following statements are equivalent:

If one (and therefore all) of these statements hold, we say that G is a semidirect product of N and H, or that G splits over N, and we write G = N⋊H.

2 Elementary facts and caveats

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.

Note that, as opposed to the case with the direct product, a semidirect product is not, in general, unique; if G and G' are two groups which both contain N as a normal subgroup and H as a subgroup, and both are a semidirect product of N and H, then it does not follow that G and G' are isomorphic.

3 Outer semidirect products

If G is a semidirect product of N and H, then the map φ : H → Aut(N) (where Aut(N) denotes the group of all automorphisms of N) defined by φ(h)(n) = hnh^-1 for all h in H and n in N is a group homomorphism. It turns out that N, H and φ together determine G up to isomorphism, as we will show next.

Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism φ : H → Aut(N) , we define a new group N ⋊_φH, the semidirect product of N and H with respect to φ as follows: the underlying set is the cartesian product N × H, and the group operation * is given by

for all n₁, n₂ in N and h₁, h₂ in H. This defines indeed a group; its identity element is (e_N, e_H) and the inverse of the element (n, h) is (φ(h^-1)(n^-1), h^-1). N × {e_H} is a normal subgroup isomorphic to N, {e_N} × H is a subgroup isomorphic to H, and the group is a semidirect product of those two subgroups in the sense given above.

Suppose now conversely that we are given an internal semidirect product as defined above, i.e. a group G with a normal subgroup N, a subgroup H, and such that every element g of G may be written uniquely in the form g=nh where n lies in N and h lies in H. Let φ : H→Aut(N) be the homomorphism

Then G is isomorphic to the outer semidirect product N_φH; the isomorphism sends the product nh to the tuple (n,h). In G, we have the rule

and this is the deeper reason for the above definition of the outer semidirect product, and an easy way to memorize it.

A version of the splitting lemma for groups states that a group G is isomorphic to a semidirect product of the two groups N and H if and only if there exists a short exact sequence

and a group homomorphism r : H → G such that v o r = id_H, the identity map on H. In this case, φ : H → Aut(N) is given by

4 Examples

The dihedral groupIn group theory, a dihedral group is a group whose elements correspond to a closed set of rotations and reflections in the plane. The dihedral group with 2''n elements is usually written as D. It is generated by a single rotation r with order n and a refl D_2n with 2n elements is isomorphic to a semidirect product of the cyclic groupIn mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that every element of the group is a power of a''. Equivalently, an element a of a gs C_n and C₂. Here, the non-identity element of C₂ acts on C_n by inverting elements; this is an automorphisms since C_n is abelianAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fo.

The group of all rigid motions of the plane (maps f : R² → R² such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in R²) is isomorphic to a semidirect product of the abelian group R² (which describes translations) and the group O(2) of orthogonal 2-by-2 matrices (which describes rotations and reflections). Every orthogonal matrix acts as an automorphism on R² by matrix multiplication.

The group O(n) of all orthogonal real n-by-n matrices (intuitively the set of all rotations and reflections of n-dimensional space) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinantIn linear algebra, the determinant is a function that associates a scalar det A to every square matrix A''. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants 1, intuitively the rotations of n-dimensional space) and C₂. If we represent C₂ as the multiplicative group of matrices {I, R}, where R is a reflection of n dimensional space (i.e. an orthogonal matrix with determinant -1), then φ : C₂ → Aut(SO(n)) is given by φ(H)(N) = H N H^-1 for all H in C₂ and N in SO(n).

5 Relation to direct products

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 which is the identity on N, 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 outer semidirect product of N and H with respect to φ(h) = id_N 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.

6 Generalizations

The construction of semidirect products can be pushed much further. There is a version in ring theory, the crossed product of rings . This is seen naturally as soon as one constructs a group ring for a semidirect product of groups. Given a group action on a topological space, there is a corresponding crossed product which will in general be non-commutative even if the group is abelian. This kind of ring can play the role of the space of orbits of the group action, in cases where that space cannot be approached by conventional topological techniques - for example in the work of Alain Connes (cf. 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.

7 See also

Wreath product

Group theory

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1 Some equivalent definitions