Dicyclic group

The dicyclic groups are special finite groups that result as an extension of cyclic groups . It is a sequence of groups of the order , Dic stands for the English name dicyclic group . ${\displaystyle \mathrm {Dic} _{n}}$ ${\displaystyle 4n}$

Construction of the group

We start from a cyclical group that we realize as a multiplicative subgroup in , ie ${\displaystyle C_{2n}}$ ${\displaystyle \mathbb {C} }$

${\displaystyle C_{2n}=\{\exp(\nu \pi i/n)|\,\nu =0,\ldots ,2n-1\}}$

The group is generated from and it is ${\displaystyle a:=\exp(\pi i/n)}$

${\displaystyle a^{\nu }=\exp(\nu \pi i/n)=\cos(\nu \pi /n)+i\sin(\nu \pi /n)}$

We're looking at the straight group order here , so that 's. By considering the complex numbers as a subalgebra of the quaternions , it is also a multiplicative subgroup of four-dimensional space . We want to add another element to the group and therefore define ${\displaystyle 2n}$ ${\displaystyle -1=a^{n}\in C_{2n}}$ ${\displaystyle \mathbb {C} =\mathbb {R} +i\mathbb {R} }$ ${\displaystyle \mathbb {H} =\mathbb {R} +i\mathbb {R} +j\mathbb {R} +k\mathbb {R} }$ ${\displaystyle C_{2n}}$ ${\displaystyle \mathbb {H} }$ ${\displaystyle b:=j}$

${\displaystyle \mathrm {Dic} _{n}:=}$of generated multiplicative subgroup of . ${\displaystyle \{a,b\}}$ ${\displaystyle \mathbb {H} }$

There is ${\displaystyle i\cdot j=k}$

${\displaystyle a^{\nu }b=j\cos(\nu \pi /n)+k\sin(\nu \pi /n)}$,

and you can show that

${\displaystyle \mathrm {Dic} _{n}=\{a^{\nu },a^{\nu }b|\,\nu =0,\ldots ,2n-1\}}$

To this one counts first and with ; This formula immediately shows that it actually only contains the specified elements. ${\displaystyle (ab)^{2}=-1}$ ${\displaystyle ba=a^{-1}b=a^{2n-1}b}$ ${\displaystyle \mathrm {Dic} _{n}}$ ${\displaystyle 4n}$

Since the elements, just like them, also span a regular 2 n corner , this group is called dicyclic , a designation that goes back to GA Miller . ${\displaystyle a^{\nu }b}$ ${\displaystyle a^{\nu }}$

The dicyclic group as an extension

The dicyclic group can be written as an extension of two cyclic groups:

${\displaystyle 0\rightarrow C_{2n}\,{\xrightarrow {\iota }}\,\mathrm {Dic_{n}} \,{\xrightarrow {p}}\,C_{2}\rightarrow 0}$.

It is the inclusion mapping and . Apparently there is a short exact sequence here . ${\displaystyle \iota }$ ${\displaystyle p(a^{\nu })=1,\,p(a^{\nu }b)=-1}$

Presentation of the dicyclic groups

With the above designations the equations obviously exist . That is enough to describe the dicyclic groups, because the dicyclic group of the order for can be obtained by the following presentation about generators and relations : ${\displaystyle a^{2n}=1,\quad a^{n}=-1=b^{2},\quad ba=a^{-1}b}$ ${\displaystyle 4n}$ ${\displaystyle n\geq 2}$

${\displaystyle \left\langle x,y\mid x^{2n}=1,x^{n}=y^{2},yx=x^{-1}y\right\rangle }$.

Dic _n for small n

${\displaystyle \mathrm {Dic} _{1}=\{1,-1,j,-j\}}$

is a group isomorphic to the cyclic group of four . ${\displaystyle C_{4}}$

${\displaystyle \mathrm {Dic} _{2}=\{1,i,-1,-i,j,ij,-j,-ij\}}$

is a group isomorphic to the quaternion group.

${\displaystyle \mathrm {Dic} _{3}=\{1,a,a^{2},a^{3},a^{4},a^{5},b,ab,a^{2}b,a^{3}b,a^{4}b,a^{5}b\}}$

is a 12-element group with the following link table :

${\displaystyle \cdot }$	1	a	a 2	a 3	a 4	a 5	b	from	a 2 b	a 3 b	a 4 b	a 5 b
1	1	a	a 2	a 3	a 4	a 5	b	from	a 2 b	a 3 b	a 4 b	a 5 b
a	a	a 2	a 3	a 4	a 5	1	from	a 2 b	a 3 b	a 4 b	a 5 b	b
a 2	a 2	a 3	a 4	a 5	1	a	a 2 b	a 3 b	a 4 b	a 5 b	b	from
a 3	a 3	a 4	a 5	1	a	a 2	a 3 b	a 4 b	a 5 b	b	from	a 2 b
a 4	a 4	a 5	1	a	a 2	a 3	a 4 b	a 5 b	b	from	a 2 b	a 3 b
a 5	a 5	1	a	a 2	a 3	a 4	a 5 b	b	from	a 2 b	a 3 b	a 4 b
b	b	a 5 b	a 4 b	a 3 b	a 2 b	from	a 3	a 2	a	1	a 5	a 4
from	from	b	a 5 b	a 4 b	a 3 b	a 2 b	a 4	a 3	a 2	a	1	a 5
a 2 b	a 2 b	from	b	a 5 b	a 4 b	a 3 b	a 5	a 4	a 3	a 2	a	1
a 3 b	a 3 b	a 2 b	from	b	a 5 b	a 4 b	1	a 5	a 4	a 3	a 2	a
a 4 b	a 4 b	a 3 b	a 2 b	from	b	a 5 b	a	1	a 5	a 4	a 3	a 2
a 5 b	a 5 b	a 4 b	a 3 b	a 2 b	from	b	a 2	a	1	a 5	a 4	a 3

Here is and . There , you can do without the powers and instead work with a sign, as we have already done with and . It is then ${\displaystyle \textstyle a={\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}}$ ${\displaystyle b=j}$ ${\displaystyle a^{3}=-1}$ ${\displaystyle a^{3},a^{4},a^{5}}$ ${\displaystyle \mathrm {Dic} _{2}}$ ${\displaystyle a=i}$ ${\displaystyle b=j}$ ${\displaystyle \mathrm {Dic} _{3}=\{1,a,a^{2},-1,-a,-a^{2},b,ab,a^{2}b,-b,-ab,-a^{2}b\}}$

Individual evidence

1. ↑ HSM Coxeter: Regular Complex Polytopes , Cambridge University Press (1974), Chapter 7.1 The Cyclic and Dicyclic groups = 74-75
2. ↑ GA Miller, HF Blichfeldt, LE Dickson: Theory and application of finite groups , New York, Wiley 1916, reprint Dover (1961)
3. ↑ Steven Roman: Fundamentals of group theory . Chapter 12, pages 347/348, Birkhäuser, Basel (2012)