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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.

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