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 following statements are equivalent:

G = NH and N ∩ H = {e} (with e being the identity element of G)
G = HN and N ∩ H = {e}
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
Every element of G can be written in one and only one way as a product of an element of H and an element of N
The natural embedding H → G, composed with the natural projection G → G/N, yields an isomorphism between H and G/N
There exists a homomorphism G → H which is the identity on H and whose kernel is N

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.

In the notation of a semidirect product the normal subgroup is written as the left factor. The common notation seems to be G = N File:Rtimes2.png H, with a cross with a vertical bar at the right (see e.g. [1]), although the names of the symbols seem to suggest that the bar should be at the side of the normal subgroup ( [2], [3]).

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. This remark leads to an extension problem, of describing the possibilities.

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 File:Rtimes2.png_φ H, 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) 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

φ(h)(n)=hnh^–1.

Then G is isomorphic to the outer semidirect product N File:Rtimes2.png_φ H; the isomorphism sends the product nh to the tuple (n,h). In G, we have the rule

(n₁h₁)(n₂h₂) = n₁(h₁n₂h₁^–1)(h₁h₂)

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

0\longrightarrow N\longrightarrow ^{\!\!\!\!\!\!\!\!\!u}\ \,G\longrightarrow ^{\!\!\!\!\!\!\!\!\!v}\ \,H\longrightarrow 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_n 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 2×2 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×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. There is also the semidirect sum of Lie algebras. 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 (see crossed product for a related construction) 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.