For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension).
, the following statements are equivalent: If any of these statements holds (and hence all of them hold, by their equivalence), we say G is the semidirect product of N and H, written or that G splits over N; one also says that G is a semidirect product of H acting on N, or even a semidirect product of H and N. To avoid ambiguity, it is advisable to specify which is the normal subgroup.
with group operation defined as The subgroups N and H determine G up to isomorphism, as we will show later.
Conversely, suppose that we are given a group G with a normal subgroup N and a subgroup H, 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 (written φ(h) = φh) given by for all n ∈ N, h ∈ H. Then G is isomorphic to the semidirect product N ⋊φ H. The isomorphism λ : G → N ⋊φ H is well defined by λ(a) = λ(nh) = (n, h) due to the uniqueness of the decomposition a = nh.
To see this, let φ be the trivial homomorphism (i.e., sending every element of H to the identity automorphism of N) then N ⋊φ H is the direct product N × H. 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 γ : H → G such that α ∘ γ = idH, the identity map on H. In this case, φ : H → Aut(N) is given by φ(h) = φh, where The dihedral group D2n with 2n elements is isomorphic to a semidirect product of the cyclic groups Cn and C2.
The presentation for this group is: More generally, a semidirect product of any two cyclic groups Cm with generator a and Cn with generator b is given by one extra relation, aba−1 = bk, with k and n coprime, and
s on the diagonal, which is called the upper unitriangular matrix group, and
If we set and then their matrix product is This gives the induced group action
The Euclidean group of all rigid motions (isometries) of the plane (maps f :
The orthogonal group O(n) of all orthogonal real n × n matrices (intuitively the set of all rotations and reflections of n-dimensional space that keep the origin fixed) 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 that keeps the origin fixed (i.e., an orthogonal matrix with determinant –1 representing an involution), then φ : C2 → Aut(SO(n)) is given by φ(H)(N) = HNH−1 for all H in C2 and N in SO(n).
In the non-trivial case (H is not the identity) this means that φ(H) is conjugation of operations by the reflection (in 3-dimensional space a rotation axis and the direction of rotation are replaced by their "mirror image").
The group of semilinear transformations on a vector space V over a field
, often denoted ΓL(V), is isomorphic to a semidirect product of the linear group GL(V) (a normal subgroup of ΓL(V)), and the automorphism group of
[6] Of course, no simple group can be expressed as a semidirect product (because they do not have nontrivial normal subgroups), but there are a few common counterexamples of groups containing a non-trivial normal subgroup that nonetheless cannot be expressed as a semidirect product.
would have to be a split extension in the following hypothetical exact sequence of groups:
This non-existence of isomorphisms can be checked by noting the trivial extension is abelian while
As opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if G and G′ are two groups that both contain isomorphic copies of 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 because the semidirect product also depends on the choice of an action of H on N. For example, there are four non-isomorphic groups of order 16 that are semidirect products of C8 and C2; in this case, C8 is necessarily a normal subgroup because it has index 2.
[8] In general, there is no known characterization (i.e., a necessary and sufficient condition) for the existence of semidirect products in groups.
For finite groups, the Schur–Zassenhaus theorem guarantees existence of a semidirect product when the order of the normal subgroup is coprime to the order of the quotient group.
Within group theory, the construction of semidirect products can be pushed much further.
The Zappa–Szép product of groups is a generalization that, in its internal version, does not assume that either subgroup is normal.
The ring-theoretic approach can be further generalized to the semidirect sum of Lie algebras.
In this context, the semidirect product is the space of orbits of the group action.
The semidirect product is a special case of the Grothendieck construction in category theory.
(respecting the group, or even just monoid structure) is the same thing as a functor from the groupoid
The semidirect product π1(X) ⋊ G is then relevant to finding the fundamental groupoid of the orbit space X/G.
Thus the existence of semidirect products reflects a failure of the category to be abelian.
Barry Simon, in his book on group representation theory,[12] employs the unusual notation
Unicode lists four variants:[13] Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice.