
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.
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.
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, 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
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
The dihedral group D_{2n} with 2n elements is isomorphic to a semidirect product of the cyclic groups C_{n} and C_{2}. Here, the nonidentity element of C_{2} 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 : R^{2} → R^{2} such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in R^{2}) is isomorphic to a semidirect product of the abelian group R^{2} (which describes translations) and the group O(2) of orthogonal 2by2 matrices (which describes rotations and reflections). Every orthogonal matrix acts as an automorphism on R^{2} by matrix multiplication.
The group O(n) of all orthogonal real nbyn matrices (intuitively the set of all rotations and reflections of ndimensional space) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of ndimensional space) and C_{2}. If we represent C_{2} 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 : C_{2} → Aut(SO(n)) is given by φ(H)(N) = H N H^{1} for all H in C_{2} and N in SO(n).
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.
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 noncommutative 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.
There are also farreaching 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.
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