Table of contents |

2 Outer semidirect products 3 Examples 4 Relation to direct products 5 Generalizations |

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.

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*_{1},*h*_{1}) * (*n*_{2},*h*_{2}) = (*n*_{1}φ(*h*_{1})(*n*_{2}),*h*_{1}*h*_{2})

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 homomorphismuv0 --->N--->G--->H---> 0

- φ(
*h*)(*n*) =*u*^{-1}(*r*(*h*)*u*(*n*)*r*(*h*^{-1})).

The dihedral group *D*_{2n} with 2*n* elements is isomorphic to a semidirect product of the cyclic groups *C*_{n} and *C*_{2}. Here, the non-identity 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 2-by-2 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 *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 *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 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.

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.