Infinite group of dieder

The infinite dihedral group is a group considered in the mathematical branch of group theory . It is a countably infinite version of the dihedral groups .

Geometric definition

Just as the dihedral groups can be introduced as the symmetry groups of a geometric figure , namely a regular n-gon , the infinite dihedral group can be defined as the group of all isometries that represent a subset of Euclidean space . is the group of all isometries that map into themselves. ${\displaystyle D_{n}}$ ${\displaystyle D_{\infty }}$ ${\displaystyle D_{\infty }}$ ${\displaystyle \mathbb {R} =\mathbb {R} ^{1}}$ ${\displaystyle \mathbb {Z} \subset \mathbb {R} }$

These isometries are translations around ${\displaystyle n}$

${\displaystyle \tau _{n}\colon \mathbb {R} \rightarrow \mathbb {R} ,\quad x\mapsto x+n}$

an integer and reflections on ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle n/2}$

${\displaystyle \sigma _{n}\colon \mathbb {R} \rightarrow \mathbb {R} ,\quad x\mapsto n-x}$

for an integer . The group of these isometries is called the infinite dihedral group . Some authors refer to this group with or after the English term "dihedral group" for dihedral group . ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle D_{\infty }}$ ${\displaystyle D_{0}}$ ${\displaystyle \mathrm {Dih} _{\infty }}$

The infinite dihedral group is already produced by and , because obviously applies ${\displaystyle \tau :=\tau _{1}}$ ${\displaystyle \sigma :=\sigma _{0}}$

${\displaystyle \tau _{n}=\tau \circ \ldots \circ \tau }$, n -fold execution for ${\displaystyle n>0}$

${\displaystyle \tau _{n}=\tau _{-n}^{-1}}$ For ${\displaystyle n<0}$

${\displaystyle \tau _{0}}$is the neutral element

${\displaystyle \sigma _{n}=\tau _{n}\circ \sigma }$for all , ${\displaystyle n\in \mathbb {Z} }$

that is, of generated sub-group contains already all isometrics and and that is, by and is generated. ${\displaystyle \{\tau ,\sigma \}}$ ${\displaystyle \tau _{n}}$ ${\displaystyle \sigma _{n}}$ ${\displaystyle D_{\infty }}$ ${\displaystyle \tau }$ ${\displaystyle \sigma }$

The relationship also exists

${\displaystyle \sigma \circ \tau \circ \sigma =\tau ^{-1}}$,

because for each applies ${\displaystyle r\in \mathbb {R} }$

${\displaystyle \sigma (\tau (\sigma (r)))=\sigma (\tau (-r))=\sigma (-r+1)=r-1=\tau ^{-1}(r)}$,

and it applies

${\displaystyle \sigma ^{2}=1}$,

where 1 denotes the neutral element, because it is a reflection. ${\displaystyle \sigma }$

D _∞ as a subgroup of the symmetry group of the circle

Let be the reflection of the unit circle at the x -axis and a rotation of the circle for an irrational number . The cyclic subgroup of the symmetry group of the circle generated by is because of the irrationality of infinite and therefore too isomorphic. Then obviously applies ${\displaystyle s}$ ${\displaystyle d}$ ${\displaystyle 2\pi r}$ ${\displaystyle r}$ ${\displaystyle d}$ ${\displaystyle r}$ ${\displaystyle \mathbb {Z} }$

${\displaystyle s^{2}=1,\,sds=d^{-1}}$

and one can show that an isomorphism from defines to the subgroup of the symmetry group of the circle generated by . In particular, their isomorphism class does not depend on the choice of the irrational number . ${\displaystyle \sigma \mapsto s,\,\tau \mapsto d}$ ${\displaystyle D_{\infty }}$ ${\displaystyle \{s,d\}}$ ${\displaystyle r}$

Presentations by D _∞

According to the above, the producers and the relations fulfill ${\displaystyle \tau }$ ${\displaystyle \sigma }$

${\displaystyle \sigma \circ \tau \circ \sigma =\tau ^{-1}}$   and   . ${\displaystyle \sigma ^{2}=1}$

One can show that there are no further independent relations. Precisely that means that the presentation ${\displaystyle D_{\infty }}$

${\displaystyle D_{\infty }\,=\,\langle x,y|\,x^{2}=1,\,xyx=y^{-1}\rangle }$

owns. The second relation can also be written as. Every product from the producers and can therefore be brought to the form with and through repeated application of the relations . The following applies to arithmetic in a group ${\displaystyle x^{2}=1}$ ${\displaystyle xy=y^{-1}x}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x^{i}y^{n}}$ ${\displaystyle i\in \{0,1\}}$ ${\displaystyle n\in \mathbb {Z} }$

${\displaystyle D_{\infty }=\{x^{i}y^{n}|\,i\in \{0,1\},n\in \mathbb {Z} \}}$   and   , ${\displaystyle x^{i}y^{n}\cdot x^{j}y^{m}=x^{i+j}y^{(1-2j)n+m}}$

where the exponent is to be understood as modulo 2. ${\displaystyle i+j}$

If you put it so is ${\displaystyle z:=xy}$

${\displaystyle z^{2}=xyxy=y^{-1}y=1}$.

Conversely, since the element can be recovered using from and back, the two involutions and , that is, elements whose square is the neutral element, are generated, and one can consider that no further relations exist. So we get a second presentation ${\displaystyle y}$ ${\displaystyle y=xz}$ ${\displaystyle x}$ ${\displaystyle z}$ ${\displaystyle D_{\infty }}$ ${\displaystyle x}$ ${\displaystyle z}$

${\displaystyle D_{\infty }=\langle x,z|\,x^{2}=1,\,z^{2}=1\rangle }$.

According to this, the infinite dihedral group is the largest group produced by two involutions, every other group is isomorphic to a factor group thereof.

Geometrically, the producer corresponds to the product , and that is the reflection on . The infinite dihedral group described geometrically above is thus also generated by the two reflections at 0 and . This is immediately understandable by realizing that the reflection at , followed by the reflection at 0, is nothing other than the translation around 1. ${\displaystyle z}$ ${\displaystyle \sigma \tau }$ ${\displaystyle \textstyle -{\frac {1}{2}}}$ ${\displaystyle \textstyle -{\frac {1}{2}}}$ ${\displaystyle \textstyle -{\frac {1}{2}}}$

D _∞ as a semi-direct product

Consider the homomorphism from the group ℤ ₂ into the automorphism group of , which maps the remainder class from 1 to . With this form the semi-direct product ${\displaystyle \alpha \colon \mathbb {Z} _{2}\rightarrow \mathrm {Aut} (\mathbb {Z} )}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \alpha _{1}\colon \mathbb {Z} \rightarrow \mathbb {Z} ,\,n\mapsto -n}$ ${\displaystyle \alpha }$

${\displaystyle \mathbb {Z} \rtimes _{\alpha }\mathbb {Z} _{2}:=\{(n,i)|\,n\in \mathbb {Z} ,i\in \{0,1\}\}}$.

The link is known by the formula

${\displaystyle (n,i)\cdot (m,j):=(n+\alpha _{i}(m),i+j)}$

defined, where and the sum is to be understood modulo 2. The isomorphism can be read from this. ${\displaystyle \alpha _{0}:=\mathrm {id} _{\mathbb {Z} }}$ ${\displaystyle i+j}$ ${\displaystyle D_{\infty }}$

Now the above is even an isomorphism, because besides there are no other nontrivial automorphisms . ${\displaystyle \alpha \colon \mathbb {Z} _{2}\rightarrow \mathrm {Aut} (\mathbb {Z} )}$ ${\displaystyle \alpha _{1}}$ ${\displaystyle \mathbb {Z} }$

Hence the holomorph of , that is ${\displaystyle D_{\infty }}$ ${\displaystyle \mathbb {Z} }$

${\displaystyle D_{\infty }\cong \mathbb {Z} \rtimes _{\alpha }\mathbb {Z} _{2}\cong \mathbb {Z} \rtimes \mathrm {Aut} (\mathbb {Z} )=\mathrm {Hol} (\mathbb {Z} )}$.

D _∞ as a free product

The infinite dihedral group is the smallest conceivable free product of nontrivial groups, it applies

${\displaystyle D_{\infty }\cong \mathbb {Z} _{2}*\mathbb {Z} _{2}}$.

It is clear that involutions are created by two. Therefore, from the above presentation, one obtains an epimorphism which is shown to be an isomorphism. Some authors define the infinite dihedral group in this way. ${\displaystyle \mathbb {Z} _{2}*\mathbb {Z} _{2}}$ ${\displaystyle D_{\infty }\rightarrow \mathbb {Z} _{2}*\mathbb {Z} _{2}}$

D _∞ as a matrix group

We look at the crowd

${\displaystyle M:={\bigl \{}{\begin{pmatrix}e&n\\0&1\end{pmatrix}};\,e\in \{-1,+1\},n\in \mathbb {Z} {\bigr \}}}$

of - matrices . The die product ${\displaystyle 2\times 2}$

${\displaystyle {\begin{pmatrix}(-1)^{i}&n\\0&1\end{pmatrix}}\cdot {\begin{pmatrix}(-1)^{j}&m\\0&1\end{pmatrix}}={\begin{pmatrix}(-1)^{i+j}&(-1)^{i}m+n\\0&1\end{pmatrix}}}$

shows that the set multiplied by the matrix product is too isomorphic a group. ${\displaystyle M}$ ${\displaystyle D_{\infty }}$

Subgroups of D _∞

The infinite dihedral group contains the following subgroups ( whole numbers): ${\displaystyle D_{\infty }=\langle x,y|\,x^{2}=1,\,xyx=y^{-1}\rangle }$ ${\displaystyle k,n,r}$

${\displaystyle U_{k}:=\langle y^{k}\rangle }$   for   , ${\displaystyle k\geq 0}$

${\displaystyle V_{0,n}:=\langle y^{n}x\rangle \cong \mathbb {Z} _{2}}$   for   , ${\displaystyle n\in \mathbb {Z} }$

${\displaystyle V_{k,r}:=\langle y^{k},y^{r}x\rangle \cong D_{\infty }}$   for   . ${\displaystyle 0\leq r<k}$

These are all subgroups of . ${\displaystyle D_{\infty }}$

Because of with , the infinite dihedral group is resolvable , even super- resolvable , metabelian and polycyclic . ${\displaystyle \{1\}\leq \langle y\rangle \leq D_{\infty }}$ ${\displaystyle D_{\infty }/\langle y\rangle \cong \mathbb {Z} _{2}}$

Individual evidence

1. ^ Wilhelm Specht : Group theory. Springer-Verlag (1956), ISBN 978-3-642-94668-4 , example 2 in section 1.2.4.
2. ^ DJS Robinson : A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , page 51: Examples of Presentations (I).
3. ^ Wilhelm Specht: Group theory. Springer-Verlag (1956), ISBN 978-3-642-94668-4 , example 1 in section 1.3.6.
4. ^ DJS Robinson: A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , Chapter 6.2: Examples of Free Products, Example II.
5. ↑ Ralph Stöcker: Algebraic Topology: An Introduction. Edition 2, Teubner-Verlag, ISBN 978-3-322-86785-8 , example 5.3.6.
6. ↑ Antonio Machi: Groups. An Introduction to Ideas and Methods of the Theory of Groups. Springer-Verlag 2012, ISBN 978-88-470-2421-2 , chapter 4.8, example 3.
7. ^ Wilhelm Specht: Group theory. Springer-Verlag (1956), ISBN 978-3-642-94668-4 , example 2 in section 1.2.4.