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Dieses Buch ist eine moderne Einführung in die Algebra, kompakt geschrieben und mit einem systematischen Aufbau. Der textual content kann für eine ein- bis zweisemestrige Vorlesung benutzt werden und deckt alle Themen ab, die für eine breite Algebra Ausbildung notwendig sind (Gruppentheorie, Ringtheorie, Körpertheorie) mit den klassischen Fragen (Quadratur des Kreises, Auflösung durch Radikale, Konstruktionen mit Zirkel und Lineal) bis zur Darstellungstheorie von endlichen Gruppen und einer Einführung in Algebren und Moduln.

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Proof: (⇒) Suppose ∼ is an equivalence relation on G. S is nonempty and thus it contains an element a. An element a is in S if and only if a ∼ e, since a = ae = ae−1 . Therefore S = e, and so for all a, b ∈ S, a ∼ e and b ∼ e so by the symmetry e ∼ b and by the transitivity of ∼, a ∼ b; thus, ab−1 ∈ S by the definition of ∼. 5 is satisfied thus S is a subgroup of G. (⇐) Let a, b and c be elements of G and suppose S is a subgroup of G. S being a subgroup implies it contains the identity e, so aa−1 = e ∈ S, thus implying a ∼ a so that ∼ is reflexive.

36 37 37 37 38 38 39 39 40 40 40 41 41 42 42 44 44 44 45 Homomorphisms. If f : G → H is a homomorphism of groups, then f (eG ) = eH and f (a−1 ) = f (a)−1 for all a ∈ G. Show by example that the first conclusion may be false if G, H are monoids that are note groups. Proof: Assuming f : G → H is a homomorphism of groups, then f (a) = f (aeG ) = f (a)f (eG ) and likewise on the left, f (a) = f (eG a) = f (eG )f (a).

The identity of H is in B, and f (e) = e, so f −1 (B) contains e and is nonempty. Take any two elements a, b ∈ f −1 (B) and using the definition let x, y ∈ B, be x = f (a) and y = f (b). 5 as applied to the subgroup B in H. 5, f −1 (B) is a subgroup of G. The nonempty subset 0 = {e} contains the identity and trivially all inverses. Therefore it is a group, and so even a subgroup of H. Thus Ker f = f −1 (0) is a subgroup of G. Certainly f |A : A → f (A) is a well-defined function (refer to Introduction, section 3).

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A Hungerford’s Algebra Solutions Manual by James Wilson


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