Semidirect product

In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal.

Some equivalent definitions

Let G be a group, N a normal subgroup of G (i.e., $N\triangleleft G$ ) 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.

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. Together N, H and φ determine G up to isomorphism, as we show now.

Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism φ : H → Aut(N), define a new group N ×_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 is 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 above.

Conversely, suppose 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 ×_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^{\!\!\!\!\!\!\!\!\!\beta}\ \, G \longrightarrow^{\!\!\!\!\!\!\!\!\!\alpha}\ \,  H \longrightarrow 0

and a group homomorphism γ : H → G such that α $\circ$ γ = id_H, the identity map on H. In this case, φ : H → Aut(N) is given by

φ(h)(n) = β⁻¹(γ(h) β(n)γ(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 automorphism since C_n is abelian. The presentation for this group is:

\langle a,\;b \mid a^2 = e,\; b^n = e,\; aba^{-1}=b^{-1}\rangle

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:

\langle a,\;b \mid a^m=e,\;b^n = e,\;aba^{-1}=b^k\;\rangle

If $r\;$ and $m\;$ are coprime, $a^r\;$ is a generator of $C_m\;$ and $a^rba^{-r}=b^{k^r}\;$ , hence the presentation:

\langle a,\;b \mid a^m=e,\;b^n = e,\;aba^{-1}=b^{k^{r}}\;\rangle

gives an group isomorphic to the previous one.

The Euclidean group of all rigid motions (isometries) 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 which keep the origin fixed). n is a translation, h a rotation or reflection. Applying a translation and then a rotation or reflection corresponds to 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). Every orthogonal matrix acts as an automorphism on R² by matrix multiplication.

The orthogonal group O(n) of all orthogonal real n×n matrices (intuitively the set of all rotations and reflections of n-dimensional 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 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 which 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) = H N H^–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 (a rotation axis and the direction of rotation are replaced by their "mirror image").

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.

Notation

Sources differ in their notation for the semidirect product. Some texts discuss it with no explicit notation. Others use the subscripted "times" symbol (×_φ) as above to modify the direct product by inclusion of a homomorphism, writing the normal group on the left. Other notation reshapes the times symbol; Unicode ^* lists four variants:

{| cellspacing="0"

! || || value || || MathML || || Unicode description ⋉ U022C9 ltimes LEFT NORMAL FACTOR SEMIDIRECT PRODUCT ⋊ U022CA rtimes RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT ⋋ U022CB lthree LEFT SEMIDIRECT PRODUCT ⋌ U022CC rthree RIGHT SEMIDIRECT PRODUCT

Although the Unicode description of the rtimes symbol says "right normal factor", a number of authors use it with a left normal factor. Therefore the usual caution for mathematical notation applies: When reading, be careful to notice the conventions adopted by the author, and when writing, explain notation choices for the reader. The choice of symbol may vary, but putting the normal factor on the left seems fairly consistent.

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