Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol Template:Math. There are two closely related concepts of semidirect product:

• an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup.
• an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and 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 (also known as splitting extension).

Inner semidirect product definitions

Given a group Template:Math with identity element Template:Math, a subgroup Template:Math, and a normal subgroup <math>N \triangleleft G</math>, the following statements are equivalent:

• Template:Math is the product of subgroups, Template:Math, and these subgroups have trivial intersection: Template:Math.
• For every Template:Math, there are unique Template:Math and Template:Math such that Template:Math.
• The composition Template:Math of the natural embedding Template:Math with the natural projection Template:Math induces an isomorphism between Template:Math and the quotient group Template:Math.
• There exists a homomorphism Template:Math that is the identity on Template:Math and whose kernel is Template:Math. In other words, there is a split exact sequence <math display="block">1 \to N \to G \to H \to 1</math> of groups (which is also known as a split extension of <math>H</math> by <math>N</math>).

If any of these statements holds (and hence all of them hold, by their equivalence), we say Template:Math is the semidirect product of Template:Math and Template:Math, written

<math>G = N \rtimes H</math> or <math>G = H \ltimes N,</math>Template:Efn

or that Template:Math splits over Template:Math; one also says that Template:Math is a semidirect product of Template:Math acting on Template:Math, or even a semidirect product of Template:Math and Template:Math. To avoid ambiguity, it is advisable to specify which is the normal subgroup.

If <math>G = N \rtimes H</math>, then there is a group homomorphism <math>\varphi : H\rightarrow \mathrm{Aut} (N)</math> given by <math>\varphi_h(n)=hnh^{-1}</math>, and for <math>g=nh,g'=n'h'</math>, we have <math>gg'=nhn'h' = nhn'h^{-1}hh' = n\varphi_{h}(n')hh' = n^* h^* </math>.

Inner and outer semidirect products

Inner semidirect product

Let us first consider the inner semidirect product. In this case, for a group <math>G</math>, consider a normal subgroup Template:Math and another subgroup Template:Math (not necessarily normal). Assume that the conditions on the list above hold. Let <math>\operatorname{Aut}(N)</math> denote the group of all automorphisms of Template:Math, which is a group under composition. Construct a group homomorphism <math>\varphi : H \to \operatorname{Aut}(N)</math> defined by conjugation,

<math>\varphi_h(n) = hnh^{-1}</math>, for all Template:Math in Template:Math and Template:Math in Template:Math.

In this way we can construct a group <math>G'=(N,H)</math> with group operation defined as

<math> (n_1, h_1) \cdot (n_2, h_2) = (n_1 \varphi_{h_1}(n_2),\, h_1 h_2)</math> for Template:Math in Template:Math and Template:Math in Template:Math.

The subgroups Template:Math and Template:Math determine Template:Math up to isomorphism, as we will show later. In this way, we can construct the group Template:Math from its subgroups. This kind of construction is called an inner semidirect product (also known as internal semidirect product<ref>DS Dummit and RM Foote (1991), Abstract algebra, Englewood Cliffs, NJ: Prentice Hall, 142.</ref>).

Outer semidirect product

Let us now consider the outer semidirect product. Given any two groups Template:Math and Template:Math and a group homomorphism Template:Math, we can construct a new group Template:Math, called the outer semidirect product of Template:Math and Template:Math with respect to Template:Math, defined as follows:<ref>Template:Cite book</ref> Template:Numbered list

This defines a group in which the identity element is Template:Math and the inverse of the element Template:Math is Template:Math. Pairs Template:Math form a normal subgroup isomorphic to Template:Math, while pairs Template:Math form a subgroup isomorphic to Template:Math. The full group is a semidirect product of those two subgroups in the sense given earlier.

Conversely, suppose that we are given a group Template:Math with a normal subgroup Template:Math and a subgroup Template:Math, such that every element Template:Math of Template:Math may be written uniquely in the form Template:Math where Template:Math lies in Template:Math and Template:Math lies in Template:Math. Let Template:Math be the homomorphism (written Template:Math) given by

<math>\varphi_h(n) = hnh^{-1}</math>

for all Template:Math.

Then Template:Math is isomorphic to the semidirect product Template:Math. The isomorphism Template:Math is well defined by Template:Math due to the uniqueness of the decomposition Template:Math.

In Template:Math, we have

<math>(n_1 h_1)(n_2 h_2) = n_1 h_1 n_2(h_1^{-1}h_1) h_2 =

   (n_1 \varphi_{h_1}(n_2))(h_1 h_2)</math>

Thus, for Template:Math and Template:Math we obtain

<math>\begin{align}

\lambda(ab) & = \lambda(n_1 h_1 n_2 h_2) = \lambda(n_1 \varphi_{h_1} (n_2) h_1 h_2) = (n_1 \varphi_{h_1} (n_2), h_1 h_2) = (n_1, h_1) \bullet (n_2, h_2) \\[5pt] & = \lambda(n_1 h_1) \bullet \lambda(n_2 h_2) = \lambda(a) \bullet \lambda(b), \end{align}</math> which proves that Template:Math is a homomorphism. Since Template:Math is obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule in Template:Math.

The direct product is a special case of the semidirect product. To see this, let Template:Math be the trivial homomorphism (i.e., sending every element of Template:Math to the identity automorphism of Template:Math) then Template:Math is the direct product Template:Math.

A version of the splitting lemma for groups states that a group Template:Math is isomorphic to a semidirect product of the two groups Template:Math and Template:Math if and only if there exists a short exact sequence

<math> 1 \longrightarrow N \,\overset{\beta}{\longrightarrow}\, G \,\overset{\alpha}{\longrightarrow}\, H \longrightarrow 1</math>

and a group homomorphism Template:Math such that Template:Math, the identity map on Template:Math. In this case, Template:Math is given by Template:Math, where

<math>\varphi_h(n) = \beta^{-1}(\gamma(h)\beta(n)\gamma(h^{-1})).</math>

Examples

Dihedral group

The dihedral group Template:Math with Template:Math elements is isomorphic to a semidirect product of the cyclic groups Template:Math and Template:Math.<ref name="mac-lane">Template:Cite book</ref> Here, the non-identity element of Template:Math acts on Template:Math by inverting elements; this is an automorphism since Template:Math is abelian. The presentation for this group is:

<math>\langle a,\;b \mid a^2 = e,\; b^n = e,\; aba^{-1} = b^{-1}\rangle.</math>

Cyclic groups

More generally, a semidirect product of any two cyclic groups Template:Math with generator Template:Math and Template:Math with generator Template:Math is given by one extra relation, Template:Math, with Template:Math and Template:Math coprime, and <math>k^m\equiv 1 \pmod{n}</math>;<ref name="mac-lane" /> that is, the presentation:<ref name="mac-lane" />

<math>\langle a,\;b \mid a^m = e,\;b^n = e,\;aba^{-1} = b^k\rangle.</math>

If Template:Math and Template:Math are coprime, Template:Math is a generator of Template:Math and Template:Math, hence the presentation:

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

gives a group isomorphic to the previous one.

Holomorph of a group

One canonical example of a group expressed as a semidirect product is the holomorph of a group. This is defined as

<math>\operatorname{Hol}(G)=G\rtimes \operatorname{Aut}(G)</math>

where <math>\text{Aut}(G)</math> is the automorphism group of a group <math>G</math> and the structure map <math>\varphi</math> comes from the right action of <math>\text{Aut}(G)</math> on <math>G</math>. In terms of multiplying elements, this gives the group structure

<math>(g,\alpha)(h,\beta)=(g(\varphi(\alpha)\cdot h),\alpha\beta).</math>

Fundamental group of the Klein bottle

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

<math>\langle a,\;b \mid aba^{-1} = b^{-1}\rangle.</math>

and is therefore a semidirect product of the group of integers with addition, <math>\mathrm{Z}</math>, with <math>\mathrm{Z}</math>. The corresponding homomorphism Template:Math is given by Template:Math.

Upper triangular matrices

The group <math>\mathbb{T}_n</math> of upper triangular matrices with non-zero determinant in an arbitrary field, that is with non-zero entries on the diagonal, has a decomposition into the semidirect product <math>\mathbb{T}_n \cong \mathbb{U}_n \rtimes \mathbb{D}_n</math><ref>Template:Cite book</ref> where <math>\mathbb{U}_n</math> is the subgroup of matrices with only <math>1</math>s on the diagonal, which is called the upper unitriangular matrix group, and <math>\mathbb{D}_n</math> is the subgroup of diagonal matrices.
The group action of <math>\mathbb{D}_n</math> on <math>\mathbb{U}_n</math> is induced by matrix multiplication. If we set

<math>A = \begin{bmatrix}

x_1 & 0 & \cdots & 0 \\ 0 & x_2 & \cdots & 0 \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & x_n \end{bmatrix}</math> and

<math>B = \begin{bmatrix}

1 & a_{12} & a_{13} & \cdots & a_{1n} \\ 0 & 1 & a_{23} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}</math> then their matrix product is

<math>AB =

\begin{bmatrix} x_1 & x_1a_{12} & x_1a_{13} & \cdots & x_1a_{1n} \\ 0 & x_2 & x_2a_{23} & \cdots & x_2a_{2n} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & x_n \end{bmatrix}.</math> This gives the induced group action <math>m:\mathbb{D}_n\times \mathbb{U}_n \to \mathbb{U}_n</math>

<math>m(A,B) = \begin{bmatrix}

1 & x_1a_{12} & x_1a_{13} & \cdots & x_1a_{1n} \\ 0 & 1 & x_2a_{23} & \cdots & x_2a_{2n} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}.</math> A matrix in <math>\mathbb{T}_n</math> can be represented by matrices in <math>\mathbb{U}_n</math> and <math>\mathbb{D}_n</math>. Hence <math>\mathbb{T}_n \cong \mathbb{U}_n \rtimes \mathbb{D}_n</math>.

Group of isometries on the plane

The Euclidean group of all rigid motions (isometries) of the plane (maps Template:Math such that the Euclidean distance between Template:Math and Template:Math equals the distance between Template:Math and Template:Math for all Template:Math and Template:Math in <math>\mathbb{R}^2</math>) is isomorphic to a semidirect product of the abelian group <math>\mathbb{R}^2</math> (which describes translations) and the group Template:Math of orthogonal Template:Math 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 Template:Math, and that the corresponding homomorphism Template:Math is given by matrix multiplication: Template:Math.

Orthogonal group O(n)

The orthogonal group Template:Math of all orthogonal real Template:Math matrices (intuitively the set of all rotations and reflections of Template:Math-dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the group Template:Math (consisting of all orthogonal matrices with determinant Template:Math, intuitively the rotations of Template:Math-dimensional space) and Template:Math. If we represent Template:Math as the multiplicative group of matrices Template:Math, where Template:Math is a reflection of Template:Math-dimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant Template:Math representing an involution), then Template:Math is given by Template:Math for all H in Template:Math and Template:Math in Template:Math. In the non-trivial case (Template:Math is not the identity) this means that Template:Math 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").

Semi-linear transformations

The group of semilinear transformations on a vector space Template:Math over a field <math>K</math>, often denoted Template:Math, is isomorphic to a semidirect product of the linear group Template:Math (a normal subgroup of Template:Math), and the automorphism group of <math>K</math>.

Crystallographic groups

In crystallography, the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group is symmorphic. Non-symmorphic space groups have point groups that are not even contained as subset of the space group, which is responsible for much of the complication in their analysis.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Dead linkTemplate:Cbignore</ref>

Non-examples

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. Note that although not every group <math>G</math> can be expressed as a split extension of <math>H</math> by <math>A</math>, it turns out that such a group can be embedded into the wreath product <math>A\wr H</math> by the universal embedding theorem.

Z₄

The cyclic group <math>\mathrm{Z}_4</math> is not a simple group since it has a subgroup of order 2, namely <math>\{0,2\} \cong \mathrm{Z}_2</math> is a subgroup and their quotient is <math>\mathrm{Z}_2</math>, so there is an extension

<math>0 \to \mathrm{Z}_2 \to \mathrm{Z}_4 \to \mathrm{Z}_2 \to 0</math>

If instead this extension is split, then the group <math>G</math> in

<math>0 \to \mathrm{Z}_2 \to G \to \mathrm{Z}_2 \to 0</math>

would be isomorphic to <math>\mathrm{Z}_2\times\mathrm{Z}_2</math>.

Q₈

The group of the eight quaternions <math>\{\pm 1,\pm i,\pm j,\pm k\}</math> where <math>ijk = -1</math> and <math>i^2 = j^2 = k^2 = -1</math>, is another example of a group<ref>{{#invoke:citation/CS1|citation |CitationClass=web

}}</ref> which has non-trivial normal subgroups yet is still not split. For example, the subgroup generated by <math>i</math> is isomorphic to <math>\mathrm{Z}_4</math> and is normal. It also has a subgroup of order <math>2</math> generated by <math>-1</math>. This would mean <math>\mathrm{Q}_8</math> would have to be a split extension in the following hypothetical exact sequence of groups:

<math>0 \to \mathrm{Z}_4 \to \mathrm{Q}_8 \to \mathrm{Z}_2 \to 0</math>,

but such an exact sequence does not exist. This can be shown by computing the first group cohomology group of <math>\mathrm{Z}_2</math> with coefficients in <math>\mathrm{Z}_4</math>, so <math>H^1(\mathrm{Z}_2,\mathrm{Z}_4) \cong \mathrm{Z}/2</math> and noting the two groups in these extensions are <math>\mathrm{Z}_2\times\mathrm{Z}_4</math> and the dihedral group <math>\mathrm{D}_8</math>. But, as neither of these groups is isomorphic with <math>\mathrm{Q}_8</math>, the quaternion group is not split. This non-existence of isomorphisms can be checked by noting the trivial extension is abelian while <math>\mathrm{Q}_8</math> is non-abelian, and noting the only normal subgroups are <math>\mathrm{Z}_2</math> and <math>\mathrm{Z}_4</math>, but <math>\mathrm{Q}_8</math> has three subgroups isomorphic to <math>\mathrm{Z}_4</math>.

Properties

If Template:Math is the semidirect product of the normal subgroup Template:Math and the subgroup Template:Math, and both Template:Math and Template:Math are finite, then the order of Template:Math equals the product of the orders of Template:Math and Template:Math. This follows from the fact that Template:Math is of the same order as the outer semidirect product of Template:Math and Template:Math, whose underlying set is the Cartesian product Template:Math.

Relation to direct products

Suppose Template:Math is a semidirect product of the normal subgroup Template:Math and the subgroup Template:Math. If Template:Math is also normal in Template:Math, or equivalently, if there exists a homomorphism Template:Math that is the identity on Template:Math with kernel Template:Math, then Template:Math is the direct product of Template:Math and Template:Math.

The direct product of two groups Template:Math and Template:Math can be thought of as the semidirect product of Template:Math and Template:Math with respect to Template:Math for all Template:Math in Template:Math.

Note that in a direct product, the order of the factors is not important, since Template:Math is isomorphic to Template:Math. 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 non-trivial homomorphism is never an abelian group, even if the factor groups are abelian.

Non-uniqueness 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 Template:Math and Template:Math are two groups that both contain isomorphic copies of Template:Math as a normal subgroup and Template:Math as a subgroup, and both are a semidirect product of Template:Math and Template:Math, then it does not follow that Template:Math and Template:Math are isomorphic because the semidirect product also depends on the choice of an action of Template:Math on Template:Math.

For example, there are four non-isomorphic groups of order 16 that are semidirect products of Template:Math and Template:Math; in this case, Template:Math is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups:

• the dihedral group of order 16
• the quasidihedral group of order 16
• the Iwasawa group of order 16

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: Template:Math.<ref name="Rose2009">Template:Cite book Note that Rose uses the opposite notation convention than the one adopted on this page (p. 152).</ref>

Existence

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In general, there is no known characterization (i.e., a 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

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.

There is also a construction in ring theory, the crossed product of rings. This is constructed in the natural way from the group ring for a semidirect product of groups. The ring-theoretic approach can be further generalized to the semidirect sum of Lie algebras.

For geometry, there is also a crossed product for group actions on a topological space; unfortunately, it is in general non-commutative even if the group is abelian. In this context, the semidirect product is the space of orbits of the group action. The latter approach has been championed by Alain Connes as a substitute for approaches by conventional topological techniques; cf. noncommutative geometry.

The semidirect product is a special case of the Grothendieck construction in category theory. Specifically, an action of <math>H</math> on <math>N</math> (respecting the group, or even just monoid structure) is the same thing as a functor

<math>F : BH \to Cat</math>

from the groupoid <math>BH</math> associated to H (having a single object *, whose endomorphisms are H) to the category of categories such that the unique object in <math>BH</math> is mapped to <math>BN</math>. The Grothendieck construction of this functor is equivalent to <math>B(H \rtimes N)</math>, the (groupoid associated to) semidirect product.<ref>Template:Harvtxt</ref>

Groupoids

Another generalization is for groupoids. This occurs in topology because if a group Template:Math acts on a space Template:Math it also acts on the fundamental groupoid Template:Math of the space. The semidirect product Template:Math is then relevant to finding the fundamental groupoid of the orbit space Template:Math. For full details see Chapter 11 of the book referenced below, and also some details in semidirect product<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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 Template:Math acting on a group Template:Math (in most cases by conjugation as subgroups of a common group) is denoted by Template:Math or Template:Math. However, some sources<ref name="Vinberg(2003)">e.g., Template:Cite book</ref> may use this symbol with the opposite meaning. In case the action Template:Math should be made explicit, one also writes Template:Math. One way of thinking about the Template:Math symbol is as a combination of the symbol for normal subgroup (Template:Math) and the symbol for the product (Template:Math). Barry Simon, in his book on group representation theory,<ref name="Simon1996">Template:Cite book</ref> employs the unusual notation <math>N\mathbin{\circledS_{\varphi}}H</math> for the semidirect product.

Unicode lists four variants:<ref>See unicode.org</ref>

	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. With the AMS symbols package loaded, \leftthreetimes produces ⋋ and \rightthreetimes produces ⋌.

See also

• Affine Lie algebra
• Grothendieck construction, a categorical construction that generalizes the semidirect product
• Holomorph
• Lie algebra semidirect sum
• Subdirect product
• Wreath product
• Zappa–Szép product
• Crossed product

References

• Template:Citation
• Template:Citation