Editing Semidirect product (section)

== 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>