Semidirect product

In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups.

Some equivalent definitions

Let G be a group, N a normal subgroup of G and H a subgroup of G.

The group G is said to be a semidirect product of N and H if G = NH and N ∩ H = {e} (with e being the identity element of G). In this case, we also say that G splits over N. We write G as N 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 both semidirect products of N and H, it does not then follow that G and G'  are isomorphic.

Equivalently, the group G is a semidirect product of N and H if every element of G can be written in one and only one way as a product of an element of N and an element of H. In particular, if both N and H are finite, then the order of G equals the product of the orders of N and H.

A third equivalent definition is the following: G is a semidirect product of N and H if the natural embedding H → G, composed with the natural projection G → G/N, provides a group isomorphism between H and G/N.

A convenient criterion is this: if H is a subgroup of G and one can find a group homomorphism f : G → H which is the identity map on H, then G is a semidirect product of the kernel of f and H. Conversely, if G is a semidirect product of N and H, then every element x of G can be written in a unique way as x = nh with n in N and h in H as mentioned above, and the assignment f(x) = h yields a group homomorphism which is the identity on H.

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:

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

(n₁, h₁) * (n₂, h₂) = (n₁ φ(h₁)(n₂), h₁ h₂)

for all n₁, n₂ in N and h₁, h₂ in H. This defines indeed a group; its identity element is (e_N, e_H). 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 that, as above, we have an internal semidirect product, so that N and H are subgroups of a group G, the subgroup N is normal in G and 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 φ(h)(n)=hnh^–1. Then G is isomorphic to the semidirect product N _φH.

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

                u      v
      0 ---> N ---> G ---> H ---> 0

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

φ(h)(n) = u^-1(r(h)u(n)r(h^-1)).

Examples

The dihedral group D_2n with 2n elements is isomorphic to a semidirect product of the cyclic groups 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 abelian.

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 determinant 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).

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.

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.