# Download Algebra by Dr. B. L. van der Waerden (auth.), Dr. B. L. van der Waerden PDF

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

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Ersetzt man nämlich in (2) a und b durch a' und b', wo ab' = ba', so folgt adb' = a' db, a d b' + beb' = a' d b + b' e b, (a d + b e) b' d = ta' d + b' e) b d, also ad Ebenso: + bc + a' d b' c b'd bd ab' = a e b' d = ac Ta ba', a' e b d, a' c = b'd· Entsprechendes gilt bei Ersetzung von (e, d) durch (e', d'), wo cd' = de' ist. 51 § 16. Quotientenbildung. Man zeigt ohne Mühe, daß alle Körpereigenschaften erfüllt sind. Das Assoziativgesetz der Addition z. B. ergibt sich so: ~+(~+~)=~+ cl+de b ( ab I d + e Cd)+-t b = dl = adl+bct+bde bdl' ad+bc +~= adl+bcl+bde bd t b dl ' und alle anderen Gesetze dementsprechend.

10), so sind 1, a, aB, ... , a",-l Repräsentanten der Nebenklassen und m ist der Index von g in

Sind in einer Menge @ Produkte äb (also Relationen der Gestalt ä b= c) definiert und ist eine Gruppe @ auf @ homomorph abgebildet, so ist auch @ eine Gruppe. Kurz: Das homomorphe Abbild einer Gruppe ist wieder eine Gruppe. Beweis. Zunächst sind je drei gegebene Elemente ä, b, c von @ stets Bilder von Elementen von @, also etwa von a, b, c. Aus folgt dann Weiter folgt aus und aus ab·c=a·bc äb·c=ä·bc. a e = a für alle a, äe = ä für alle a, ba =e ba =e. e Also gibt es in @ ein Einselement und zu jedem ä ein Inverses.