Editing Semidirect product

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{{Short description|Operation in group theory}}
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In [[mathematics]], specifically in [[group theory]], the concept of a '''semidirect product''' is a generalization of a [[direct product of groups|direct product]]. It is usually denoted with the symbol {{math|⋉}}. There are two closely related concepts of semidirect product: 
* an ''inner'' semidirect product is a particular way in which a [[Group (mathematics)|group]] can be made up of two [[subgroup]]s, 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 group]]s, the [[Schur–Zassenhaus theorem]] provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as [[Group_extension#Classifying_split_extensions | splitting extension]]).

== Inner semidirect product definitions ==

Given a group {{math|''G''}} with [[identity element]] {{math|''e''}}, a [[subgroup]] {{math|''H''}}, and a [[normal subgroup]] <math>N \triangleleft G</math>, the following statements are equivalent:
* {{math|''G''}} is the [[product of group subsets#Product of subgroups|product of subgroups]], {{math|1=''G'' = ''NH''}}, and these subgroups have trivial intersection: {{math|1=''N'' ∩ ''H'' = {{mset|''e''}}}}.
* For every {{math|''g'' ∈ ''G''}}, there are unique {{math|''n'' ∈ ''N''}} and {{math|''h'' ∈ ''H''}} such that {{math|1=''g'' = ''nh''}}.
* The [[function composition|composition]] {{math|''π'' ∘ ''i''}} of the natural embedding {{math|''i'' : ''H'' → ''G''}} with the natural projection {{math|''π'' : ''G'' → ''G''/''N''}} induces an [[group isomorphism|isomorphism]] between {{math|''H''}} and the [[quotient group]] {{math|''G''/''N''}}.
* There exists a [[group homomorphism|homomorphism]] {{math|''G'' → ''H''}} that is the [[identity function|identity]] [[restriction (mathematics)|on]] {{math|''H''}} and whose [[kernel (algebra)|kernel]] is {{math|''N''}}. 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 [[Group_extension#Classifying_split_extensions | 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 {{math|''G''}} is the '''semidirect product''' of {{math|''N''}} and {{math|''H''}}, written
: <math>G = N \rtimes H</math> or <math>G = H \ltimes N,</math>{{efn|The symbol <math>\rtimes</math> is a combination of <math>\triangleleft</math> and <math>\times</math>, oriented so that <math>N \triangleleft (N \rtimes H)</math>.<ref>{{cite web |last1=Neumann |first1=Walter |author-link1=Walter Neumann |title=Notes on semidirect products |page=3 |url=https://www.math.columbia.edu/~bayer/S09/ModernAlgebra/semidirect.pdf#page=3 |access-date=30 December 2024 |archive-url=http://web.archive.org/web/20240716064845/https://www.math.columbia.edu/~bayer/S09/ModernAlgebra/semidirect.pdf#page=3 |archive-date=16 July 2024}}</ref>
}}
or that {{math|''G''}} ''splits'' over {{math|''N''}}; one also says that {{math|''G''}} is a '''semidirect''' product of {{math|''H''}} acting on {{math|''N''}}, or even a semidirect product of {{math|''H''}} and {{math|''N''}}. 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  {{math|''N''}} and another subgroup {{math|''H''}} (not necessarily normal). Assume that the
conditions on the list above hold. Let <math>\operatorname{Aut}(N)</math> denote the group of all [[Automorphism group|automorphism]]s of {{math|''N''}}, 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 {{math|''h''}} in {{math|''H''}} and {{math|''n''}} in {{math|''N''}}. 
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 {{math|''n''<sub>1</sub>, ''n''<sub>2</sub>}} in {{math|''N''}} and {{math|''h''<sub>1</sub>, ''h''<sub>2</sub>}} in {{math|''H''}}. 
The subgroups {{math|''N''}} and {{math|''H''}} determine {{math|''G''}} [[up to]] isomorphism, as we will show later. In this way, we can construct the group {{math|''G''}} 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 {{math|''N''}} and {{math|''H''}} and a group homomorphism {{math|''φ'' : ''H'' → Aut(''N'')}}, we can construct a new group {{math|''N'' ⋊{{sub|''φ''}} ''H''}}, called the '''outer semidirect product''' of {{math|''N''}} and {{math|''H''}} with respect to {{math|''φ''}}, defined as follows:<ref>{{cite book |last1=Robinson |first1=Derek John Scott |title=An Introduction to Abstract Algebra |year=2003 |publisher=[[De Gruyter|Walter de Gruyter]] |isbn=9783110175448 |pages=75–76}}</ref>
{{numbered list
| The underlying set is the [[Cartesian product]] {{math|''N'' × ''H''}}.
| The group operation <math>\bullet</math> is determined by the homomorphism {{math|''φ''}}:
: <math>\begin{align}
  \bullet : (N \rtimes_\varphi H) \times (N \rtimes_\varphi H) &\to N \rtimes_\varphi H\\
                                         (n_1, h_1) \bullet (n_2, h_2) &= (n_1 \varphi_{h_1}(n_2),\, h_1 h_2)
\end{align}</math>
for {{math|''n''<sub>1</sub>, ''n''<sub>2</sub>}} in {{math|''N''}} and {{math|''h''<sub>1</sub>, ''h''<sub>2</sub>}} in {{math|''H''}}.
}}

This defines a group in which the identity element is {{math|(''e''<sub>''N''</sub>, ''e<sub>H</sub>'')}} and the inverse of the element {{math|(''n'', ''h'')}} is {{math|(''φ''<sub>''h''<sup>−1</sup></sub>(''n''<sup>−1</sup>), ''h''<sup>−1</sup>)}}. Pairs {{math|(''n'', ''e<sub>H</sub>'')}} form a normal subgroup isomorphic to {{math|''N''}}, while pairs {{math|(''e<sub>N</sub>'', ''h'')}} form a subgroup isomorphic to {{math|''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 {{math|''G''}} with a normal subgroup {{math|''N''}} and a subgroup {{math|''H''}}, such that every element {{math|''g''}} of {{math|''G''}} may be written uniquely in the form {{math|''g {{=}} nh''}} where {{math|''n''}} lies in {{math|''N''}} and {{math|''h''}} lies in {{math|''H''}}. Let {{math|''φ'' : ''H'' → Aut(''N'')}} be the homomorphism (written {{math|''φ''(''h'') {{=}} ''φ''<sub>''h''</sub>}}) given by
: <math>\varphi_h(n) = hnh^{-1}</math>
for all {{math|''n'' ∈ ''N'', ''h'' ∈ ''H''}}.

Then {{math|''G''}} is isomorphic to the semidirect product {{math|''N'' ⋊{{sub|''φ''}} ''H''}}. The isomorphism {{math|''λ'' : ''G'' → ''N'' ⋊{{sub|''φ''}} ''H''}} is well defined
by {{math|''λ''(''a'') {{=}} ''λ''(''nh'') {{=}} (''n, h'')}} due to the uniqueness of the decomposition {{math|''a'' {{=}} ''nh''}}.

In {{math|''G''}}, 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 {{math|''a'' {{=}} ''n''{{sub|1}}''h''{{sub|1}}}} and {{math|''b'' {{=}} ''n''{{sub|2}}''h''{{sub|2}}}} 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 [[mathematical proof|proves]] that {{math|''λ''}} is a homomorphism. Since {{math|''λ''}} is obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule in {{math|''N'' ⋊{{sub|''φ''}} ''H''}}.

The direct product is a special case of the semidirect product. To see this, let {{math|''φ''}} be the trivial homomorphism (i.e., sending every element of {{math|''H''}} to the identity automorphism of {{math|''N''}}) then {{math|''N'' ⋊{{sub|''φ''}} ''H''}} is the direct product {{math|''N'' × ''H''}}.

A version of the [[splitting lemma]] for groups states that a group {{math|''G''}} is isomorphic to a semidirect product of the two groups {{math|''N''}} and {{math|''H''}} [[if and only if]] there exists a [[Exact sequence#Short exact sequence|short exact sequence]]
: <math> 1 \longrightarrow N \,\overset{\beta}{\longrightarrow}\, G \,\overset{\alpha}{\longrightarrow}\, H \longrightarrow 1</math>
and a group homomorphism {{math|''γ'' : ''H'' → ''G''}} such that {{math|''α''&thinsp;∘&thinsp;''γ'' {{=}} id<sub>''H''</sub>}}, the identity map on {{math|''H''}}. In this case, {{math|''φ'' : ''H'' → Aut(''N'')}} is given by {{math|''φ''(''h'') {{=}} ''φ''<sub>''h''</sub>}}, where
:<math>\varphi_h(n) = \beta^{-1}(\gamma(h)\beta(n)\gamma(h^{-1})).</math>

== Examples ==

=== Dihedral group ===
The [[dihedral group]] {{math|D{{sub|2''n''}}}} with {{math|2''n''}} elements is isomorphic to a semidirect product of the [[cyclic group]]s {{math|C{{sub|''n''}}}} and {{math|C{{sub|2}}}}.<ref name="mac-lane">{{cite book |last1=Mac Lane |first1=Saunders |author-link1=Saunders Mac Lane |last2=Birkhoff |first2=Garrett |author-link2=Garrett Birkhoff |title=Algebra |edition=3rd |year=1999 |publisher=American Mathematical Society |isbn=0-8218-1646-2 |pages=414–415}}</ref> Here, the non-identity element of {{math|C{{sub|2}}}} acts on {{math|C{{sub|''n''}}}} by inverting elements; this is an automorphism since {{math|C{{sub|''n''}}}} is [[abelian group|abelian]]. The [[group presentation|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 {{math|C{{sub|''m''}}}} with generator {{math|''a''}} and {{math|C{{sub|''n''}}}} with generator {{math|''b''}} is given by one extra relation, {{math|''aba''{{sup|−1}} {{=}} ''b{{sup|k}}''}}, with {{math|''k''}} and {{math|''n''}} [[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 {{math|''r''}} and {{math|''m''}} are coprime, {{math|''a{{sup|r}}''}} is a generator of {{math|C{{sub|''m''}}}} and {{math|''a{{sup|r}}ba{{sup|−r}}'' {{=}} ''b{{sup|k{{sup|r}}}}''}}, 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 (mathematics)|holomorph]] of a group. This is defined as<blockquote><math>\operatorname{Hol}(G)=G\rtimes \operatorname{Aut}(G)</math></blockquote>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<blockquote><math>(g,\alpha)(h,\beta)=(g(\varphi(\alpha)\cdot h),\alpha\beta).</math></blockquote>

=== 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 {{math|''φ'' : <math>\mathrm{Z}</math> → Aut(<math>\mathrm{Z}</math>)}} is given by {{math|''φ''(''h'')(''n'') {{=}} (−1){{sup|''h''}}''n''}}.

=== Upper triangular matrices ===
The group <math>\mathbb{T}_n</math> of upper [[triangular matrix|triangular matrices]] with non-zero [[determinant]] in an arbitrary field, that is with non-zero entries on the [[Main diagonal|diagonal]], has a decomposition into the semidirect product
<math>\mathbb{T}_n \cong \mathbb{U}_n \rtimes \mathbb{D}_n</math><ref>{{Cite book|last=Milne|url=https://www.jmilne.org/math/CourseNotes/iAG200.pdf |archive-url=https://web.archive.org/web/20160307074150/http://www.jmilne.org/math/CourseNotes/iAG200.pdf |archive-date=2016-03-07 |url-status=live|title=Algebraic Groups|pages=45, semi-direct products}}</ref> where <math>\mathbb{U}_n</math> is the subgroup of [[matrix (mathematics)|matrices]] with only <math>1</math>s on the diagonal, which is called the upper [[Triangular matrix|unitriangular matrix]] group, and <math>\mathbb{D}_n</math> is the subgroup of [[Diagonal matrix|diagonal matrices]].<br />
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 multiplication|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 ([[isometry|isometries]]) of the plane (maps {{math|''f'' : <math>\mathbb{R}</math>{{sup|2}} → <math>\mathbb{R}</math>{{sup|2}}}} such that the Euclidean distance between {{math|''x''}} and {{math|''y''}} equals the distance between {{math|''f''(''x'')}} and {{math|''f''(''y'')}} for all {{math|''x''}} and {{math|''y''}} 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 {{math|O(2)}} of [[orthogonal matrix|orthogonal]] {{math|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 [[Conjugation of isometries in Euclidean space|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 {{math|O(2)}}, and that the corresponding homomorphism {{math|''φ'' : O(2) → Aut(<math>\mathbb{R}</math>{{sup|2}})}} is given by [[matrix multiplication]]: {{math|''φ''(''h'')(''n'') {{=}} ''hn''}}.

=== Orthogonal group O(''n'') ===
The [[orthogonal group]] {{math|O(''n'')}} of all orthogonal [[real number|real]] {{math|''n'' × ''n''}} matrices (intuitively the set of all rotations and reflections of {{math|''n''}}-dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the group {{math|SO(''n'')}} (consisting of all orthogonal matrices with [[determinant]] {{math|1}}, intuitively the rotations of {{math|''n''}}-dimensional space) and {{math|C{{sub|2}}}}. If we represent {{math|C{{sub|2}}}} as the multiplicative group of matrices {{math|{''I'', ''R''}{{null}}}}, where {{math|''R''}} is a reflection of {{math|''n''}}-dimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant {{math|–1}} representing an [[involution (mathematics)|involution]]), then {{math|''φ'' : C{{sub|2}} → Aut(SO(''n''))}} is given by {{math|''φ''(''H'')(''N'') {{=}} ''HNH''{{sup|−1}}}} for all ''H'' in {{math|C{{sub|2}}}} and {{math|''N''}} in {{math|SO(''n'')}}. In the non-trivial case ({{math|''H''}} is not the identity) this means that {{math|''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").

=== Semi-linear transformations ===
The group of [[semilinear transformation]]s on a vector space {{math|''V''}} over a field <math>K</math>, often denoted {{math|ΓL(''V'')}}, is isomorphic to a semidirect product of the [[linear group]] {{math|GL(''V'')}} (a [[normal subgroup]] of {{math|ΓL(''V'')}}), 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 [[wikt: symmorphic|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>{{cite web|last1=Thompson|first1=Nick|title=Irreducible Brillouin Zones and Band Structures|url=https://bandgap.io/blog/brillouin_zones/|website=bandgap.io|access-date=13 December 2017}}{{dead link|date=March 2025|bot=medic}}{{cbignore|bot=medic}}</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]].

=== Z<sub>4</sub> ===
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 <blockquote><math>0 \to \mathrm{Z}_2 \to \mathrm{Z}_4 \to \mathrm{Z}_2 \to 0</math></blockquote>If instead this extension is [[Split extension|split]], then the group <math>G</math> in<blockquote><math>0 \to \mathrm{Z}_2 \to G \to \mathrm{Z}_2 \to 0</math></blockquote>would be isomorphic to <math>\mathrm{Z}_2\times\mathrm{Z}_2</math>.

=== Q<sub>8</sub> ===
The [[Quaternion group|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>{{Cite web|title=abstract algebra - Can every non-simple group $G$ be written as a semidirect product?|url=https://math.stackexchange.com/q/1504422 |access-date=2020-10-29|website=Mathematics Stack Exchange}}</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: <blockquote><math>0 \to \mathrm{Z}_4 \to \mathrm{Q}_8 \to \mathrm{Z}_2 \to 0</math>, </blockquote>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 {{math|''G''}} is the semidirect product of the normal subgroup {{math|''N''}} and the subgroup {{math|''H''}}, and both {{math|''N''}} and {{math|''H''}} are finite, then the [[order of a group|order]] of {{math|''G''}} equals the product of the orders of {{math|''N''}} and {{math|''H''}}. This follows from the fact that {{math|''G''}} is of the same order as the outer semidirect product of {{math|''N''}} and {{math|''H''}}, whose underlying set is the [[Cartesian product]] {{math|''N'' × ''H''}}.

=== Relation to direct products ===
Suppose {{math|''G''}} is a semidirect product of the normal subgroup {{math|''N''}} and the subgroup {{math|''H''}}. If {{math|''H''}} is also normal in {{math|''G''}}, or equivalently, if there exists a homomorphism {{math|''G'' → ''N''}} that is the identity on {{math|''N''}} with kernel {{math|''H''}}, then {{math|''G''}} is the [[direct product of groups|direct product]] of {{math|''N''}} and {{math|''H''}}.

The direct product of two groups {{math|''N''}} and {{math|''H''}} can be thought of as the semidirect product of {{math|''N''}} and {{math|''H''}} with respect to {{math|''φ''(''h'') {{=}} id{{sub|''N''}}}} for all {{math|''h''}} in {{math|''H''}}.

Note that in a direct product, the order of the factors is not important, since {{math|''N'' × ''H''}} is isomorphic to {{math|''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 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 of groups|direct product]], a semidirect product of two groups is not, in general, unique; if {{math|''G''}} and {{math|''G′''}} are two groups that both contain isomorphic copies of {{math|''N''}} as a normal subgroup and {{math|''H''}} as a subgroup, and both are a semidirect product of {{math|''N''}} and {{math|''H''}}, then it does ''not'' follow that {{math|''G''}} and {{math|''G′''}} are [[group isomorphism|isomorphic]] because the semidirect product also depends on the choice of an action of {{math|''H''}} on {{math|''N''}}.

For example, there are four non-isomorphic groups of order 16 that are semidirect products of {{math|C{{sub|8}}}} and {{math|C{{sub|2}}}}; in this case, {{math|C{{sub|8}}}} 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: {{math|(D{{sub|8}} ⋉ C{{sub|3}}) ≅ (C{{sub|2}} ⋉ [[Dicyclic group|Q{{sub|12}}]]) ≅ (C{{sub|2}} ⋉ D{{sub|12}}) ≅ (D{{sub|6}} ⋉ [[Klein four-group|V]])}}.<ref name="Rose2009">{{cite book|author=H.E. Rose|title=A Course on Finite Groups|year=2009|publisher=Springer Science & Business Media|isbn=978-1-84882-889-6|page=183}} Note that Rose uses the opposite notation convention than the one adopted on this page (p. 152).</ref>

=== Existence ===
{{main|Schur–Zassenhaus theorem}}
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 (group theory)|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|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 [[Lie algebra extension#By semidirect sum|semidirect sum of Lie algebras]].  

For geometry, there is also a crossed product for [[Group action (mathematics)|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>{{harvtxt|Barr|Wells|2012|loc=§12.2}}</ref>

=== Groupoids ===

Another generalization is for groupoids. This occurs in topology because if a group {{math|''G''}} acts on a space {{math|''X''}} it also acts on the [[fundamental groupoid]] {{math|''π''{{sub|1}}(''X'')}} of the space.  The semidirect product {{math|''π''{{sub|1}}(''X'') ⋊ ''G''}} is then relevant to finding the fundamental groupoid of the [[orbit space]] {{math|''X/G''}}. For full  details see Chapter 11 of the book referenced below, and also some details in semidirect product<ref>{{cite web| url = http://ncatlab.org/nlab/show/semidirect+product| title = Ncatlab.org}}</ref> in [[nLab|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 {{math|''H''}} acting on a group {{math|''N''}} (in most cases by conjugation as subgroups of a common group) is denoted by {{math|''N'' ⋊ ''H''}} or {{math|''H'' ⋉ ''N''}}. However, some sources<ref name="Vinberg(2003)">e.g., {{cite book|author=E. B. Vinberg|title=A Course in Algebra|location=Providence, RI|publisher=American Mathematical Society|page=389|isbn=0-8218-3413-4|date=2003}}</ref> may use this symbol with the opposite meaning. In case the action {{math|''φ'' : ''H'' → Aut(''N'')}} should be made explicit, one also writes {{math|''N'' ⋊{{sub|''φ''}} ''H''}}. One way of thinking about the {{math|''N'' ⋊ ''H''}} symbol is as a combination of the symbol for normal subgroup ({{math|◁}}) and the symbol for the product ({{math|×}}). [[Barry Simon]], in his book on group representation theory,<ref name="Simon1996">{{cite book|author=B. Simon|title=Representations of Finite and Compact Groups|location=Providence, RI|publisher=American Mathematical Society|page=6|isbn=0-8218-0453-7|date=1996}}</ref> employs the unusual notation <math>N\mathbin{\circledS_{\varphi}}H</math> for the semidirect product.

[[Unicode]] lists four variants:<ref>See [https://www.unicode.org/charts/#symbols unicode.org]</ref>

: {| class="wikitable"
!   !! 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 (mathematics)|Holomorph]]
* [[Lie algebra extension#By semidirect sum|Lie algebra semidirect sum]]
* [[Subdirect product]]
* [[Wreath product]]
* [[Zappa–Szép product]]
* [[Crossed product]]

== Notes ==
{{notelist}}
{{reflist}}

== References ==
{{refimprove|date=June 2009}}
* {{citation
 | last1       = Barr
 | first1      = Michael
 | last2       = Wells
 | first2      = Charles
 | title       = Category theory for computing science
 | series = Reprints in Theory and Applications of Categories
 | volume      = 2012
 | issue       = 22
 | pages       = 558
 | year        = 2012
 | language    = English
 | zbl = 1253.18001
}}
* {{citation
 | last       = Brown
 | first      = R.
 | title      = Topology and groupoids
 | publisher  = Booksurge
 | year       = 2006
 | isbn       = 1-4196-2722-8
}}

[[Category:Group products]]