By Joël Schmidt

"Que chacun cherche à être utile à lui-même et aux autres". De sa naissance à los angeles veille de sa mort, Johann Wolfgang von Goethe (1749-1832) n'a cessé d'écrire. L'édition complète de ses oeuvres compte cent cinquante volumes. Comme d'aucuns l'affirment, aspect de vie où l'oeuvre à réaliser ait tenu rôle plus capital. Romans, poèmes épiques, oeuvres scientifiques, livrets d'opéra, dessins, théories de l'art, pièces de théâtre, Goethe s'essaie à tous les genres. Sa soif d'expériences est insatiable : biologie, zoologie, ostéologie, optique, géologie. Grand administrateur et homme d'Etat, amoureux infatigable, l'auteur des Souffrances du jeune Werther, de Faust, des Affinités électives mais aussi d'un fameux Traité des couleurs nous est ici dévoilé dans sa vie l. a. plus quotidienne, dépoussiéré, dégagé de toutes ses légendes.

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**Example text**

Solutions to Chapter 1 Let G be partitioned by the set {H X a I a EA} of cosets of H, and let H be partitioned by the set {KY{3 I f3 E B} of cosets of K. Suppose that g E G. Then we have g E HX a for some a E A and so g = hX a for some (unique) h E H. But h E KY{3 for some f3 E B and so we have that g = kY{3x a for some k E K. Thus we see that every element of G belongs to a coset K Y{3x a for some f3 E B and some a E A. The result now follows from the fact that if KY{3x a = KY{3'x a , then, since the left hand side is contained in the coset H X a and the right hand side is contained in the coset H X a ', we have necessarily X a = X a ', which gives KY{3 = Kyf3' and hence Y{3 = Y{3" Now observe that (HnK)x = HxnKx for all subgroups Hand K of G.

But h E KY{3 for some f3 E B and so we have that g = kY{3x a for some k E K. Thus we see that every element of G belongs to a coset K Y{3x a for some f3 E B and some a E A. The result now follows from the fact that if KY{3x a = KY{3'x a , then, since the left hand side is contained in the coset H X a and the right hand side is contained in the coset H X a ', we have necessarily X a = X a ', which gives KY{3 = Kyf3' and hence Y{3 = Y{3" Now observe that (HnK)x = HxnKx for all subgroups Hand K of G.

Since it is clearly surjective, it follows by the first isomorphism theorem that D 2n is a quotient group of D oo . 12 = a. (i) Define f: ([+ -> IR+ by f(a+ib) which is surjective. Since Then f is a group morphism the result follows by the first isomorphism theorem. (ii) Define f : ([" -> U by a+i b -> Then f a . b ~+~~. ~ va 2 = I} ~ IR;o. The result now follows by the first isomorphism theorem. l if a> OJ if a < O. Then f is a surjective group morphism with Ker f = IR;o' Also, define 9 : IQ" -> G2 by if a> 0; if a < O.