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By Martin Goldstern

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Example text

2 Beispiel. K = {1, 2}, A1 = (A1 , ·, e, −1 ), A2 = (A2 , +, 0, −) seien Gruppen. Dann wird in A1 × A2 = (A1 × A2 , ◦, (e, 0), ) folgendermaßen gerechnet: (a1 , a2 ) ◦ (b1 , b2 ) = (a1 b1 , a2 + b2 ), 41 (a1 , a2 ) = (a−1 1 , −a2 ). Es gilt: A1 ×A2 ist eine Gruppe. Assoziativgesetz: ((a1 , a2 )◦(b1 , b2 ))◦(c1 , c2 ) = (a1 b1 c1 , a2 + b2 + c2 ) = (a1 , a2 ) ◦ ((b1 , b2 ) ◦ (c1 , c2 )); (e, 0) ist neutrales Element: (e, 0) ◦ (a1 , a2 ) = (ea1 , 0 + a2 ) = (a1 , a2 ) = (a1 e, a2 + 0) = (a1 , a2 ) ◦ (e, 0); (a1 , a2 ) ist Inverses von (a1 , a2 ): (a1 , a2 ) ◦ −1 (a1 , a2 ) = (a1 , a2 ) ◦ (a−1 1 , −a2 ) = (a1 a1 , a2 + (−a2 )) = (e, 0), analog (a1 , a2 ) ◦ (a1 , a2 ) = (e, 0).

Es ist klar, dass N ∩ U Untergruppe von G (und ebenso von N und U ) ist. 3. F¨ ur alle g ∈ G, somit erst recht f¨ ur alle g ∈ N U , gilt g −1 N g = N , daher ist N ⊳ N U . 4. F¨ ur u ∈ U , n ∈ N ist u−1 nu ∈ u−1 N u = N . Wenn n u ¨berdies noch in u liegt, gilt −1 u nu ∈ N ∩ U . Daher gilt N ∩ U ⊳ U . 5. Sei f : G → G/N die kanonische Abbildung. Wegen U ⊆ N U ist die Identit¨at idU auf U ein Homomorphismus von U nach N U . Die Abbildung h := f ◦ idU : U → N U/N ist surjektiv auf N U/N , denn f¨ ur jede Klasse (nu)N gilt (nu)N = uN = h(u).

Wenn wir also eine Funktion f durch eine Vorschrift (∗) definieren, m¨ ussen wir uns immer erst vergewissern, dass (∗∗) erf¨ ullt ist. 14 Definition. Die so erhaltene Algebra A/π := (A/π, (ωi∗ )i∈I ) heißt Faktoralgebra von A nach der Kongruenz π. Oft setzt man ωi := ωi∗ . 15 Beispiel. A = (Z, +, 0, −, ·, 1), π = ≡ mod n. Die Faktoralgebra A/π ist dann gegeben durch (Zn , +∗ , 0∗ , −∗ , ·∗ , 1∗ ) mit [a] +∗ [b] = [a + b], 0∗ = [0], −∗ [a] = [−a], [a] ·∗ [b] = [ab], 1∗ = [1] (d. , man rechnet mit den Repr¨asentanten der Klassen).

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