Semidirect product

Concepts in group theory
category of groups
subgroups, normal subgroups
group homomorphisms, kernel, image, quotient
direct product, direct sum
semidirect product, wreath product
Types of groups
simple, finite, infinite
discrete, continuous
multiplicative, additive
cyclic, abelian, dihedral
nilpotent, solvable
list of group theory topics
glossary of group theory

In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product. It is a cartesian product as a set, but with a particular multiplication operation.

Some equivalent definitions

Let G be a group with identity element e, N a normal subgroup of G (i.e., N ◁ G) and H a subgroup of G. The following statements are equivalent:

• G = NH and N ∩ H = {e}.
• G = HN and N ∩ H = {e}.
• Every element of G can be written as a unique product of an element of N and an element of H.
• Every element of G can be written as a unique 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 the quotient group 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, written $G = N \rtimes H,$ 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. In order to avoid ambiguities, it is advisable to specify which of the two subgroups is normal.

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 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. This remark leads to an extension problem, of describing the possibilities.

Semidirect products and group homomorphisms

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, where 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), there is a new group $N\rtimes_{\varphi}H$ (or simply $N\times_{\varphi}H$), called the semidirect product of N and H with respect to φ, defined as follows.

• As a set, $N\rtimes_{\varphi}H$ is the cartesian product N × H.
• Multiplication of elements in $N\rtimes_{\varphi}H$ is determined by the homomorphism φ. The operation is

$*\colon (N\rtimes_{\varphi} H)\times(N\rtimes_{\varphi} H)\to N\rtimes_{\varphi} H$

defined by

$(n_1, h_1)*(n_2, h_2) = (n_1\varphi_{h_1}(n_2), h_1h_2)$

for n₁, n₂ in N and h₁, h₂ in H.

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 above.

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

φ_h(n) = hnh ^{− 1}

for all n in N and h in H. Then G is isomorphic to the semidirect product $N\rtimes_{\varphi}H$ ; the isomorphism sends the product nh to the tuple (n,h). In G, we have the multiplication rule

$(n_1,h_1)(n_2,h_2) = (n_1(h_1n_2h_1^{-1}),h_1h_2).$

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

$1\longrightarrow N \longrightarrow^{\!\!\!\!\!\!\!\!\!\beta}\ \, G \longrightarrow^{\!\!\!\!\!\!\!\!\!\alpha}\ \, H \longrightarrow 1$

and a group homomorphism γ : H → G such that $\alpha \circ \gamma = \operatorname{id}_H$, the identity map on H. In this case, φ : H → Aut(N) is given by φ(h) = φ_h, where

φ_h(n) = β ^{− 1}(γ(h)β(n)γ(h ^{− 1})).

If φ is the trivial homomorphism, sending every element of H to the identity automorphism of N, then $N\rtimes_{\varphi}H$ is the direct product $N \times H$.

Examples

The dihedral group D_2n 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 a group isomorphic to the previous one.

The fundamental group of the Klein bottle can be presented in the form

$\langle a,\;b \mid aba^{-1}=b^{-1}\;\rangle$

and is therefore a semidirect product of the group of integers, $\mathbb{Z}$, with itself.

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. 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 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.

Groupoids

Another generalisation is for groupoids. This occurs in topology because if a group G acts on a space X it also acts on the fundamental groupoid π₁(X) of the space. The semidirect product $\pi_1(X) \rtimes G$ 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 in ncatlab.

Abelian categories

Non-trivial 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 $N\rtimes H$ or $H\ltimes N$. However, some sources may use this symbol with the opposite meaning. In case the action $\phi :H \rightarrow \operatorname{Aut}(N)$ should be made explicit, one also writes $N\rtimes_{\phi}H$. One way of thinking about the $N\rtimes H$ symbol is as a combination of the symbol for normal subgroup ($\triangleleft$) and the symbol for the product ($\times$).

Unicode lists four variants:

		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

• Holomorph
• Subdirect product
• Wreath product

References

• R. Brown, Topology and groupoids, Booksurge 2006. ISBN 1-4196-2722-8