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By Hans-Jürgen Schneider

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Außerdem ist (xa)−1 = a−1 x−1 = x−1 a ∈ HN mit einem a ∈ N . Betrachte nun die Abbildung ϕ : H → HN/N , x → x. ϕ ist Epimorphismus mit ker ϕ = H ∩ N , und die Behauptung folgt damit aus dem Homomorphiesatz. Beispiele. 1. Es sei n 2. Das Vorzeichen sign : Sn → Z/2Z = {±1} ist ein Gruppenepimorphismus, dessen Kern An := ker sign man die alternierende Gruppe nennt. Nach dem Homomorphiesatz ist Sn /An ∼ = {±1}. 2. Für n 1 ist die Abbildung Z → µn (C) = {z ∈ C | z n = 1}, a → ζ a mit ζ := e2πi/n ein Epimorphismus von Gruppen.

1870 erscheint das erste Lehrbuch über Galois- und Gruppentheorie, verfaßt von Camille Jordan. 14 Literatur zu Galois: T. Rothman, Genius and Biographers: The Fictionalization of Evariste Galois, Am. Math. T. Rigatelli, Evariste Galois 1811-1832, Birkhäuser 1996. 1 Restklassenbildung bei Gruppen Definition. Es sei G eine Gruppe. 1. Ist M ⊂ G eine Teilmenge, so heißt M := {xε11 . . xεmm | m ∈ N, x1 , . . , xm ∈ M, ε1 , . . , εm ∈ {±1}} die von M erzeugte Untergruppe von G. Für g1 , . . , gn ∈ G setzt man zur Einsparung von Klammern g1 , .

Dies ist äquivalent zu g −1 hg ∈ Stab(x), also zu h ∈ g Stab(x)g −1 . 2. Ist H ⊂ G eine Untergruppe, so sind die Bahnen bei der Rechtsoperation G × H → G durch Translation (also g • h := gh, wobei rechts das Produkt in G steht) gerade die Linksnebenklassen xH, x ∈ G. Definition. Es sei G eine Gruppe und µ : G × G → G, (g, x) → gxg −1 die Operation von G auf sich selbst durch Konjugation. h. x ∼ y genau dann, wenn es ein g ∈ G gibt mit y = gxg −1 ; man sagt dann auch, x und y seien konjugiert. 1.

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