Download Algebra und Zahlentheorie [Lecture notes] by Walter Gubler PDF

By Walter Gubler

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Additional info for Algebra und Zahlentheorie [Lecture notes]

Example text

G/N heißt Faktorgruppe . 5. Es gibt eine kanonische Projektion mit π : G → G/N, x → x ¯ := π(x) := xN = [x] Da wir in G/N repr¨ asentantenweise rechnen, folgt unmittelbar, dass π ein surjektiver Gruppenhomomorphismus ist. Weiterhin ist ker(π) = N . Dies muß allerdings noch separat bewiesen werden. Beweis: g ∈ ker(π) ⇔ gN = g¯ = e¯ = eN = N . Damit ist g = g · e ∈ N . Umgekehrt folgt aus g ∈ N aber auch gN = N . 6. Sei ϕ : G1 → G2 ein Gruppenhomomorphismus. Dann ist ker(ϕ) G1 . 5 haben wir gesehen, dass ker(ϕ) eine Untergruppe von G1 ist.

Hierbei bilden die Kongruenzklassen modulo m den Ring1 Z/mZ. Wir sagen ”a ∈ Z ist kongruent zu b ∈ Z modulo mZ” genau dann, wenn gilt: a ≡ b (mod m) :⇔ m|a − b ⇔ a − b ∈ mZ ⇔ −b + a ∈ mZ Das Ziel der beiden folgenden Abschnitte ist es, dies f¨ ur beliebige H und G zu verallgemeinern. Wir wollen also ganz allgemein lernen in ”G modulo H” zu rechnen. 1. Wir lassen uns durch das Obige leiten und definieren analog: a ∼ b (mod H) :⇔ b−1 · a ∈ H f¨ ur a, b ∈ G. Alternativ h¨ atte man an dieser Stelle auch die Definition a ∼ b (mod H) :⇔ a · b−1 ∈ H ableiten k¨onnen.

2. NEBENKLASSEN 41 Den Fall einer endlichen Ordnung haben wir damit also gezeigt. Zu pr¨ ufen ist nun nur noch der Fall ord(g) = ∞. Nach unserer anf¨ang¨ lichen Uberlegung folgt aus g n = g m immer g n−m = e und damit, da ord(g) = ∞, in diesem Fall n = m. h. unendlich viele und die Behauptung ist somit gezeigt. 14. 15 (Satz von Euler). Seien ord(G) < ∞ und g ∈ G. Dann ist g ord(G) = e. 8 folgt ord(< g >)| ord(G). 13, dass ord(g) = ord(< g >). Damit gibt es ein k ∈ N mit k · ord(g) = ord(G), so dass g ord(G) = g k·ord(g) = (g ord(g) )k = ek = e gilt.