Quasi-Diedergruppe

In der Mathematik sind Quasi-Diedergruppen gewisse endliche nicht-abelsche Gruppen der Ordnung ${\displaystyle 2^{n}}$, wobei ${\displaystyle n\geq 4}$ ist.

Definition

Eine Quasi-Diedergruppe ist eine Gruppe, die von zwei Elementen ${\displaystyle a}$ und ${\displaystyle b}$ der Form

${\displaystyle \langle a,b\mid a^{2^{n-1}}=b^{2}=1,bab=a^{2^{n-2}-1}\rangle }$

mit ${\displaystyle n\geq 4}$ erzeugt wird.

Anzahl Elemente

Aus ${\displaystyle bab=a^{2^{n-2}-1}}$ folgt wegen ${\displaystyle b^{2}=1}$, dass ${\displaystyle ba=a^{2^{n-2}-1}b}$. Also kann jedes endliche Produkt der Erzeuger ${\displaystyle a}$ und ${\displaystyle b}$ der Quasi-Diedergruppe durch Anwendung dieser Regel auf die Form ${\displaystyle a^{i}b^{j}}$ gebracht werden. Wegen ${\displaystyle a^{2^{n-1}}=b^{2}=1}$ folgt:

Die Quasi-Diedergruppe hat 2ⁿ Elemente: ${\displaystyle \{1,a,a^{2},\ldots ,a^{2^{n-1}},b,ba,ba^{2},\ldots ,ba^{2^{n-1}}\}}$

Beispiel

Die kleinste Quasi-Diedergruppe hat die Ordnung ${\displaystyle 16}$ und wird von zwei Elementen ${\displaystyle a}$ und ${\displaystyle b}$ erzeugt, die die Gleichungen ${\displaystyle a^{8}=b^{2}=1}$ und ${\displaystyle bab=a^{3}}$ erfüllen. Da ${\displaystyle b^{2}=1}$, folgt aus der letzten Gleichung nach Rechtsmultiplikation mit ${\displaystyle b}$, dass ${\displaystyle ba=a^{3}b}$. Also kann man in einer beliebigen Folge von ${\displaystyle a}$'s und ${\displaystyle b}$'s jedes vor einem ${\displaystyle a}$ stehende ${\displaystyle b}$ hinter das ${\displaystyle a}$ bringen, wenn man dieses durch ${\displaystyle a^{3}}$ ersetzt. Daraus folgt dann, dass alle Elemente dieser Gruppe von der Form ${\displaystyle 1,a,a^{2},\ldots ,a^{7},b,ab,\ldots ,a^{7}b}$ sind. Ferner lassen sich mit obigen Gleichungen sämtliche Multiplikationen in der Gruppe bestimmen. Als Beispiel betrachten wir die beiden Produkte aus ${\displaystyle a^{2}}$ und ${\displaystyle a^{3}b}$:

${\displaystyle a^{2}\cdot a^{3}b=a^{5}b}$     (denn ${\displaystyle a^{2}a^{3}=a^{5}}$)

${\displaystyle a^{3}b\cdot a^{2}=a^{3}a^{3}ba=a^{3}a^{3}a^{3}b=a^{9}b=ab}$     (zweimal ${\displaystyle b}$ nach rechts bringen und ${\displaystyle a^{8}=1}$ verwenden)

Insgesamt erhalten w​ir die folgende Verknüpfungstafel

${\displaystyle \,\cdot }$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$
${\displaystyle \,1}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$
${\displaystyle \,a}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$
${\displaystyle \,a^{2}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$
${\displaystyle \,a^{3}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$
${\displaystyle \,a^{4}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$
${\displaystyle \,a^{5}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$
${\displaystyle \,a^{6}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$
${\displaystyle \,a^{7}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,1}$	${\displaystyle \,a}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{6}b}$
${\displaystyle \,b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$
${\displaystyle \,ab}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$
${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$
${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$
${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$
${\displaystyle \,a^{5}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$
${\displaystyle \,a^{6}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$
${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{7}b}$	${\displaystyle \,a^{2}b}$	${\displaystyle \,a^{5}b}$	${\displaystyle \,b}$	${\displaystyle \,a^{3}b}$	${\displaystyle \,a^{6}b}$	${\displaystyle \,ab}$	${\displaystyle \,a^{4}b}$	${\displaystyle \,a^{7}}$	${\displaystyle \,a^{2}}$	${\displaystyle \,a^{5}}$	${\displaystyle \,1}$	${\displaystyle \,a^{3}}$	${\displaystyle \,a^{6}}$	${\displaystyle \,a}$	${\displaystyle \,a^{4}}$

Siehe auch

• Diedergruppe
• Liste kleiner Gruppen

Literatur

• Bertram Huppert: Endliche Gruppen (= Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Bd. 134, ISSN 0072-7830). Band 1. Springer, Berlin u. a. 1967, S. 90–93.