Read e-book online A Radical Approach to Algebra (Addison-Wesley Series in PDF

By Mary Gray

ISBN-10: 020102568X

ISBN-13: 9780201025682

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Extra info for A Radical Approach to Algebra (Addison-Wesley Series in Mathematics)

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Das Einselement bezeichnet man in diesem Fall mit 0, weil es, genau so wie die Null im Bereich der ganzen Zahlen, durch die Eigenschaft + O+a=a charakterisiert ist. Analog wird in einem Modul das inverse Element von a mit -a bezeichnet. Für a + (-b) schreibt man kurz a -b, weil dieses Element die Lösung der Gleichung x + b = a ist: (a - b) + b = a + (- b + b) = a + °= a. Gruppen. 24 Aufgaben. 1. Die euklidischen Bewegungen und Umlegungen des Raumes (d. h. diejenigen Transformationen, bei denen alle Entfernungen der Punktepaare ungeändert bleiben) bilden eine unendliche nichtabelsche Gruppe.

A) Assoziatives Gesetz: a' b c = ab' c. IH. Distributivgesetze. a) a' (b + c) = ab + a c. b) (b + c) . a = ba + c a. Zusatz. Gilt auch für die Multiplikation das kommutative Gesetz: II. b) a· b = b . a, so heißt der Ring kommutativ. Vorläufig werden wir es hauptsächlich mit kommutativen Ringen zu tun haben. Zu den Gesetzen der Addition. Die drei Gesetze Ia, b, c zusammen besagen nichts anderes, als daß die Ringelemente bei der Addition eine abelsche Gruppe bilden 2. Also können wir alle früher 1 2 Eindeutige Lösbarkeit wird nicht verlangt, folgt aber später.

4. Die symmetrische Gruppe 6 4 hat außer sich selbst und der Einheitsgruppe nur die folgenden Normalteiler: a) die alternierende Gruppe 2(" b) die "KLEINsche Vierergruppe" j8" bestehend aus den Permutationen (1), (12) (34), (13) (24), (14) (23). Die letztere Gruppe ist abelsch und isomorph zu der in Aufgabe 1 abstrakt definierten Gruppe. 5. l) eine "Zwischengruppe" : so ist gauch Normalteiler in Sj. 6. Alle unendlichen zyklischen Gruppen sind isomorph zur additiven Gruppe der ganzen Zahlen. 7. Die mit einem Element a vertauschbaren Elemente x in einer Gruppe

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A Radical Approach to Algebra (Addison-Wesley Series in Mathematics) by Mary Gray

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