Algebra I by Dr. B. L. van der Waerden (auth.)

By Dr. B. L. van der Waerden (auth.)

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Additional resources for Algebra I

Sample text

Stellt man jede ganze Zahl m in der Form m=qn+r dar, so ist am = aqn+r = aqnar = (an)q ar (0 ~ r = ear = < n) ar. Also sind alle Potenzen von a schon in der Reihe ao, a1, ... , an-1 vertreten. Die zyklische Gruppe hat demnach genau n Elemente, nämlich aOa1, ... ,an-1. Die Zahl n, die Ordnung der von a erzeugten zyklischen Gruppe, heißt die Ordnung des Elements a. Sind alle Potenzen von a verschieden, so nennt man a ein Element unendlicher Ordnung. Beispiele. Die ganzen Zahlen ... ' - 2, - 1, 0, 1, 2, ...

Ist g eine Untergruppe und a ein Element von@, so bezeichnet man den Komplex ag als eine linksseitige Nebenklasse, den Komplex gaalseine rechtsseitigeNebenklasse (auch N ebengruppe, Nebenkomplex oder Restklasse) von g in @. Liegt a in g, so ist a g = g, also ist stets eine der linksseitigen (und ebenso eine der rechtsseitigen) Nebenklassen von g gleich g selbst. Im folgenden werden hauptsächlich die linksseitigen Nebenklassen betrachtet, obwohl alle anzustellenden Betrachtungen auch für die rechtsseitigen Nebenklassen gelten.

Und 4. besagen, daß in g das Einselement liegt und daß g mit a auch das Untergruppen 21 inverse Element a-1 enthält. Davon ist wieder die Forderung des Einselements überflüssig; denn wenn a irgendein Element von g ist, so liegt in g auch a-1, also auch das Produkt aa-l= e. Damit ist bewiesen: Notwendig und hinreichend, damit eine nichtleere Untermenge g einer gegebenen Gruppe @ eine Untergruppe ist, sind die folgenden Bedingungen: 1. g enthält mit je zwei Elementen a, b auch das Produkt ab; 2. g enthält zu jedem Element a auch das inverse Element a-1.

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