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1 1 1 0 bop 324 548 a Fh(Mathematisc)m(hes)37 b(Institut)1652 b(SS)37 b(2003)324 668 y(der)g(Univ)m(ersit)972 673 y(\177)973 668 y(at)f(M)1216 673 y(\177)1213 668 y(unc)m(hen)1657 b(Blatt)36 b(1)324 789 y(Prof.)h(Dr.)g(B.)g(P)m(areigis)1637 1287 y Fg(Algebra)45 b(I)t(I)443 1515 y Ff(1.)k(Sei)37 b Fe(G)h Ff(eine)f(endlic)m(he)h(ab)s(elsc)m(he)g(Grupp)s(e)g(und)g Fe(S)43 b Ff(:=)36 b Fd(f)p Fe(z)41 b Fd(2)c Fc(C)62 b Fd(j)36 b(j)p Fe(z)t Fd(j)h Ff(=)f(1)p Fd(g)h Ff(der)568 1635 y(Einheitskreis,)28 b(b)s(etrac)m(h)m(tet)i(als)e(m)m(ultiplik)-5 b(ativ)24 b(gesc)m(hrieb)s(ene)31 b(Grupp)s(e.)e(Die)e(zu)568 1756 y Fe(G)32 b Ff(duale)g(Grupp)s(e)1736 1851 y(^)1714 1876 y Fe(G)c Ff(:=)g(Hom)o(\()p Fe(G;)17 b(S)6 b Ff(\))568 2050 y(ist)26 b(mit)f(den)j(punkt)m(w)m(eisen)h(Op)s(erationen)d(eine)h (ab)s(elsc)m(he)h(Grupp)s(e.)f(Zeigen)f(Sie,)568 2170 y(da\031)36 b Fe(G)g Ff(und)1090 2145 y(^)1068 2170 y Fe(G)h Ff(isomorph)e(sind.)h(\(Hin)m(w)m(eis:)h(V)-8 b(erw)m(enden)39 b(Sie)d(den)h(Hauptsatz)568 2291 y(f)601 2295 y(\177)598 2291 y(ur)d(endlic)m(he)g(ab)s(elsc)m(he)g(Grupp)s(en;) h(b)s(etrac)m(h)m(ten)g(Sie)f(zun)2723 2295 y(\177)2723 2291 y(ac)m(hst)i(den)e(F)-8 b(all)32 b(einer)568 2411 y(zyklisc)m(hen)i(Grupp)s(e.\))2017 b(\(6\))443 2614 y(2.)49 b(Sei)32 b Fe(G)g Ff(eine)h(endlic)m(he)f(ab)s(elsc)m(he)i (Grupp)s(e)e(mit)f(der)i(dualen)g(Grupp)s(e)3191 2589 y(^)3169 2614 y Fe(G)p Ff(.)609 2818 y(\(a\))48 b(Zeigen)32 b(Sie,)g(da\031)h(es)g(zu)g(jedem)g Fe(g)e Fd(2)d Fe(G)k Ff(ein)g Fe(\015)h Fd(2)2627 2793 y Ff(^)2605 2818 y Fe(G)g Ff(mit)d Fe(\015)5 b Ff(\()p Fe(g)t Ff(\))27 b Fd(6)p Ff(=)h(1)k(gibt.)603 2979 y(\(b\))49 b(Zeigen)32 b(Sie,)g(da\031)h(die)f(Abbildung)1659 3232 y Fe(G)27 b Fd(!)1913 3181 y Ff(^)1913 3207 y(^)1890 3232 y Fe(G;)17 b(g)31 b Fd(7!)c Ff(\()p Fe(\015)33 b Fd(7!)27 b Fe(\015)5 b Ff(\()p Fe(g)t Ff(\)\))782 3452 y(ein)32 b(Grupp)s(enisomorphism)m (us)g(ist.)1329 b(\(4\))443 3655 y(3.)49 b(Seien)24 b Fe(m)g Ff(und)g Fe(n)h Ff(zw)m(ei)f(nat)1541 3659 y(\177)1538 3655 y(urlic)m(he)f(Zahlen.)h(Zeigen)f(Sie,)g(da\031)h(die)g(Grupp)s (en)g Fc(Z)3455 3670 y Fb(mn)568 3776 y Ff(und)g Fc(Z)823 3791 y Fb(m)890 3776 y Fd(\002)t Fc(Z)1040 3791 y Fb(n)1108 3776 y Ff(genau)g(dann)g(isomorph)e(sind,)h(w)m(enn)i Fe(m)f Ff(und)g Fe(n)g Ff(teilerfremd)e(sind.)3437 3896 y(\(5\))443 4099 y(4.)49 b(Seien)29 b Fe(a)870 4114 y Fa(0)910 4099 y Fe(;)17 b(a)1005 4114 y Fa(1)1044 4099 y Fe(;)g(a)1139 4114 y Fa(2)1178 4099 y Fe(;)g(a)1273 4114 y Fa(3)1313 4099 y Fe(;)g(a)1408 4114 y Fa(4)1476 4099 y Ff(ungerade)30 b(ganze)f(Zahlen.)g(Zeigen)f(Sie,)h(da\031)g(das) g(P)m(oly-)568 4220 y(nom)1428 4340 y Fe(a)1479 4355 y Fa(0)1541 4340 y Ff(+)22 b Fe(a)1690 4355 y Fa(1)1729 4340 y Fe(x)h Ff(+)f Fe(a)1956 4355 y Fa(2)1995 4340 y Fe(x)2050 4299 y Fa(2)2112 4340 y Ff(+)g Fe(a)2261 4355 y Fa(3)2301 4340 y Fe(x)2356 4299 y Fa(3)2418 4340 y Ff(+)g Fe(a)2567 4355 y Fa(4)2607 4340 y Fe(x)2662 4299 y Fa(4)570 4518 y Ff(\177)568 4514 y(ub)s(er)33 b Fc(Q)50 b Ff(irreduzib)s(el)31 b(ist.)1918 b(\(5\))324 4742 y(Abgab)s(e:)32 b(Dienstag,)g(14.4.2003,)f(11.00)h(Uhr,)2088 4721 y(\177)2076 4742 y(Ubungsk)2449 4746 y(\177)2449 4742 y(asten)i(im)d(1.)h(Sto)s(c)m(k)i(des)f(Ma-)324 4862 y(thematisc)m(hen)g(Instituts.)324 4983 y(Bitte)f(geb)s(en)h(Sie)f (auf)g(Ihrer)h(L)1475 4987 y(\177)1475 4983 y(osung)g(Ihren)h(Namen)e (und)h(Ihre)2753 4961 y(\177)2741 4983 y(Ubungsgrupp)s(e)h(an.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF