New PDF release: Algebra I

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

ISBN-10: 3540035613

ISBN-13: 9783540035619

ISBN-10: 3642960448

ISBN-13: 9783642960444

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Jedem Element a entspricht eine Restklasse a, und diese Zuordnung ist homomorph, da der Summe a + b die Summe a + b und dem Produkt ab ebenso ab entspricht. Also bilden die Restklassen einen Ring (§ 12). Diesen Ring bezeichnen wir als den Restklassenring o/m von 0 nach dem Ideal m oder von 0 modulo m. Der Ring 0 ist auf o/m mittels des angegebenen Zuordnungsverfahrens homo1 AIle Kongruenzen natiirlich modulo m. 52 Ringe und Korper morph abgebildet. Das Ideal m spielt bei diesem Homomorphismus genau die Rolle des friiheren n.

M, m 28 Gruppen Die Automorphismen einer Menge bringen gewissermaBen ihre Symmetrieeigenschaften zum Ausdruck. B. ) in sich iibergeht, wobei gewisse Relationen (Entfemungen, Winkel, Lagebeziehungen) erhalten bleiben, oder in unserer Terminologie, daB die Figur in bezug auf ihre metrischen Eigenschaften gewisse Automorphismen gestattet. Offenbar ist das Produkt zweier Automorphismen (Produktbildung von Transformationen nach § 6) wieder ein Automorphismus und die inverse Operation eines Automorphismus wieder ein solcher.

Die ganzen Zahlen ... , - 2, - 1, 0, 1, 2, ... mit der Addition als Verkniipfung bilden eine unendliche zyklische Gruppe. Die oben angeschriebenen Gruppen @)2, m:a sind zyklische Gruppen der Ordnungen 2, 3. Aufgaben. 1. Es gibt zyk1ische Permutationsgruppen beliebiger Ordnung. 2. Man beweise durch Induktion nach n, daB die n - 1 Transpositionen (12). •• , (1 n) fur n > 1 die symmetrische Gruppe IS.. erzeugen. 3. Ebenso, daB die n - 2 Dreierzyklen (1 2 3), (1 24), ... , (1 2n) fiir n > 2 die alternierende Gruppe lll..

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

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