Semidirect product
Algebraic structure → Group theory Group theory 


Modular groups

Infinite dimensional Lie group

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: an inner semidirect product is a particular way in which a group can be constructed from two subgroups, one of which is a normal subgroup, while an outer semidirect product is a Cartesian product as a set, but with a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products.
For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (aka split[ting] extension).
Inner semidirect product definitions
Given a group G with identity element e, a subgroup H, and a normal subgroup N ◁ G; then the following statements are equivalent:
 G is the product of subgroups, G = NH, where the subgroups have trivial intersection, N ∩ H = {e}.
 For every g ∈ G, there are unique n ∈ N and h ∈ H, such that g = nh.
 For every g ∈ G, there are unique h ∈ H and n ∈ N, such that g = hn.
 The composition π ∘ i of the natural embedding i: H → G and the natural projection π: G → G / N is an isomorphism between H and the quotient group G / N.
 There exists a homomorphism G → H that is the identity on H and whose kernel is N.
If these statements hold, 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.
Outer semidirect products
Let G be a semidirect product of the normal subgroup N and the subgroup H. Let Aut(N) denote the group of all automorphisms of N. The map φ: H → Aut(N) defined by φ(h) = φ_{h}, conjugation by h, where φ(h)(n) = φ_{h}(n) = hnh^{−1} for all h in H and n in N, is a group homomorphism. (Note that hnh^{−1} ∈ N since N is normal in G.) Together N, H, and φ determine G up to isomorphism, as we show now.
Given any (even unrelated) two groups N and H and a group homomorphism φ: H → Aut(N), we can construct a new group N ⋊_{φ} H, called the (outer) semidirect product of N and H with respect to φ, defined as follows.^{[1]}
 The underlying set is the Cartesian product N × H.
 The operation, ∙, is determined by the homomorphism, φ:
This defines a group in which the identity element is (e_{N}, e_{H}) and the inverse of the element (n, h) is (φ_{h−1}(n^{−1}), h^{−1}). Pairs (n, e_{H}) form a normal subgroup isomorphic to N, while pairs (e_{N}, h) form a subgroup isomorphic to H. The full group is a semidirect product of those two subgroups in the sense given earlier.
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 given by φ(h) = φ_{h}, where
for all n ∈ N, h ∈ H.
Then G is isomorphic to the semidirect product N ⋊_{φ} H; and applying the isomorphism to the product, nh, gives the tuple, (n, h). In G, we have
which shows that the above map is indeed an isomorphism and also explains the definition of the multiplication rule in N ⋊_{φ} H.
The direct product is a special case of the semidirect product. 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 = id_{H}, the identity map on H. In this case, φ: H → Aut(N) is given by φ(h) = φ_{h}, where
Examples
The dihedral group D_{2n} with 2n elements is isomorphic to a semidirect product of the cyclic groups C_{n} and C_{2}.^{[2]} Here, the nonidentity element of C_{2} acts on C_{n} by inverting elements; this is an automorphism since C_{n} is abelian. The presentation for this group is:
More generally, a semidirect product of any two cyclic groups C_{m} with generator a and C_{n} with generator b is given by a single relation aba^{−1} = b^{k} with k and n coprime; i.e., the presentation:^{[2]}
If r and m are coprime, a_{r} is a generator of C_{m} and a^{r}ba^{−r} = b^{kr}, hence the presentation:
gives a group isomorphic to the previous one.
The fundamental group of the Klein bottle can be presented in the form
and is therefore a semidirect product of the group of integers, ℤ, with ℤ. The corresponding homomorphism φ: ℤ → Aut(ℤ) is given by φ(h)(n) = (−1)^{h}n.
The Euclidean group of all rigid motions (isometries) 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 × 2 matrices (which describes rotations and reflections that keep the origin fixed). Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the conjugate of the original translation). This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and O(2), and that the corresponding homomorphism φ: O(2) → Aut(ℝ^{2}) is given by matrix multiplication: φ(h)(n) = hn.
The orthogonal group O(n) of all orthogonal real n × n matrices (intuitively the set of all rotations and reflections of ndimensional space which 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 ndimensional space) and C_{2}. If we represent C_{2} as the multiplicative group of matrices {I, R}, where R is a reflection of ndimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant –1 representing an involution), then φ: C_{2} → Aut(SO(n)) is given by φ(H)(N) = HNH^{−1} for all H in C_{2} and N in SO(n). In the nontrivial case (H is not the identity) this means that φ(H) is conjugation of operations by the reflection (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 𝕂, 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 𝕂.
Properties
As a consequence of Lagrange's theorem, 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.
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 that 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 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.
Furthermore, the result of a (proper) semidirect product by means of a nontrivial homomorphism is never an abelian group, even if the factor groups are abelian.
Nonuniqueness of semidirect products (and further examples)
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 which 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 nonisomorphic groups of order 16 that are [semi]direct product of C_{8} and C_{2}; C_{8} is necessarily a normal subgroup in this case because it has index 2 in a group of order 16. One of these four [semi]direct products is the direct product, while the other three are nonabelian groups:
 the dihedral group of order 16
 the quasidihedral group of order 16
 the Iwasawa group of order 16 (M16)
For comparative Cayley diagrams of these see Example 3 in .
If a given group is a semidirect product, then there is no guarantee that this decomposition is unique. For example, there is a group of order 24 (the only one containing six elements of order 4 and six elements of order 6) that can be expressed as semidirect product in the following ways: (D_{8} ⋉ C_{3}) ≃ (C_{2} ⋉ Q_{12}) ≃ (C_{2} ⋉ D_{12}) ≃ (D_{6} ⋉ V).^{[3]}
Existence
In general, there's no known characterization (necessary and sufficient condition) for the existence of semidirect products in groups. However, some sufficient conditions are known, which guarantee existence in certain cases. 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.
For example, the Schur–Zassenhaus theorem implies the existence of a semidirect product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance.
Generalizations
The construction of semidirect products can be pushed much further. The Zappa–Szep product of groups is a generalization which, in its internal version, does not assume that either subgroup is normal. There is also a construction 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 noncommutative 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 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.
Groupoids
Another generalization is for groupoids. This occurs in topology because if a group G acts on a space X it also acts on the fundamental groupoid π_{1}(X) of the space. The semidirect product π_{1}(X) ⋊ G is then relevant to finding the fundamental groupoid of the orbit space X/G. For full details see Chapter 11 of the book referenced below, and also some details in semidirect product^{[4]} in ncatlab.
Abelian categories
Nontrivial semidirect products do not arise in abelian categories, such as the category of modules. In this case, the splitting lemma shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.
Notation
Usually the semidirect product of a group H acting on a group N (in most cases by conjugation as subgroups of a common group) is denoted by N ⋊ H or H ⋉ N. However, some sources may use this symbol with the opposite meaning. In case the action φ: H → Aut(N) should be made explicit, one also writes N ⋊_{φ} H. One way of thinking about the N ⋊ H symbol is as a combination of the symbol for normal subgroup (◁) and the symbol for the product (×).
Unicode lists four variants:^{[5]}
Value MathML Unicode description ⋉ U+22C9 ltimes LEFT NORMAL FACTOR SEMIDIRECT PRODUCT ⋊ U+22CA rtimes RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT ⋋ U+22CB lthree LEFT SEMIDIRECT PRODUCT ⋌ U+22CC rthree RIGHT SEMIDIRECT PRODUCT
Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice.
In LaTeX, the commands \rtimes and \ltimes produce the corresponding characters.
See also
Notes
 ↑ Robinson, Derek John Scott (2003). An Introduction to Abstract Algebra. Walter de Gruyter. pp. 75–76. ISBN 9783110175448.
 1 2 Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3 ed.). American Mathematical Society. pp. 414–415. ISBN 0821816462.
 ↑ H.E. Rose (2009). A Course on Finite Groups. Springer Science & Business Media. p. 183. ISBN 9781848828896. Note that Rose uses the opposite notation convention than the one adopted on this page (p. 152).
 ↑ Ncatlab.org
 ↑ See unicode.org
References
 R. Brown, Topology and groupoids, Booksurge 2006. ISBN 1419627228