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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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/FontInfo 7 dict dup begin
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/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
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currentdict end
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%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
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/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
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readonly def
/FontBBox{-24 -250 1110 750}readonly def
currentdict end
currentfile eexec
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%%BeginFont: LINE10
%!PS-AdobeFont-1.1: LINE10 1.001
%%CreationDate: 1992 Oct 23 20:22:05
%%RevisionDate: 2001 Jun 05 20:22:05
% Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.001) readonly def
/Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def
/FullName (LINE10) readonly def
/FamilyName (LaTeX) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /LINE10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
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dup 63 /a63 put
dup 64 /a64 put
dup 82 /a82 put
readonly def
/FontBBox{-150 -150 1020 1020}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 33 /arrowright put
dup 50 /element put
dup 54 /negationslash put
dup 80 /P put
dup 86 /V put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 84 /T put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 122 /z put
dup 127 /dieresis put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 19 /iota put
dup 23 /nu put
dup 59 /comma put
dup 61 /slash put
dup 65 /A put
dup 66 /B put
dup 77 /M put
dup 81 /Q put
dup 84 /T put
dup 86 /V put
dup 88 /X put
dup 89 /Y put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 102 /f put
dup 107 /k put
dup 110 /n put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 22 /macron put
dup 25 /germandbls put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 53 /five put
dup 54 /six put
dup 58 /colon put
dup 61 /equal put
dup 63 /question put
dup 65 /A put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 77 /M put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 90 /Z put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 122 /z put
dup 127 /dieresis put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 69 /E put
dup 75 /K put
dup 85 /U put
dup 97 /a put
dup 98 /b put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 116 /t put
dup 117 /u put
dup 122 /z put
dup 127 /dieresis put
readonly def
/FontBBox{-33 -250 945 749}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 54 /six put
dup 65 /A put
dup 66 /B put
dup 68 /D put
dup 73 /I put
dup 77 /M put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 127 /dieresis put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
currentdict end
currentfile eexec
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TeXDict begin 1 0 bop 115 355 a Fj(Mathematisc)m(hes)39
b(Institut)100 471 y(der)33 b(Univ)m(ersit)742 476 y(\177)743
471 y(at)h(M)984 476 y(\177)981 471 y(unc)m(hen)206 588
y(Prof.)j(Dr.)g(B.)g(P)m(areigis)3104 354 y(SS)h(2000)3124
471 y(Blatt)f(6)413 1084 y Fi(\177)396 1116 y(Ubungen)45
b(zu)f(Einf)1374 1121 y(\177)1370 1116 y(uhrung)g(in)g(die)f
(Kategorien)l(theorie)0 1370 y Fj(Aufgab)s(e)c(21.)190
b Fh(\(1\))41 b(Ein)d(Monoid)f Fg(M)49 b Fh(hei\031t)37
b Ff(r)-5 b(e)g(chts)39 b(k)2207 1374 y(\177)2207 1370
y(urzb)-5 b(ar)p Fh(,)38 b(w)m(enn)g(f)2825 1374 y(\177)2822
1370 y(ur)g(alle)f Fg(a;)17 b(b;)g(c)36 b Fe(2)g Fg(M)315
1487 y Fh(gilt:)31 b(aus)g Fg(ac)d Fh(=)g Fg(bc)j Fh(folgt)f
Fg(a)e Fh(=)g Fg(b)p Fh(.)j(Jede)i(Grupp)s(e)e(k)-5 b(ann)31
b(als)g(rec)m(h)m(ts)i(und)f(links)g(k)3244 1491 y(\177)3241
1487 y(urzbares)315 1603 y(Monoid)g(angesehen)j(w)m(erden.)315
1719 y(Zeigen)26 b(Sie,)i(da\031)e(es)h(zu)g(jedem)h(k)m(omm)m(utativ)m
(en)h(k)2175 1723 y(\177)2172 1719 y(urzbaren)f(Monoid)e
Fg(M)38 b Fh(eine)27 b(ab)s(elsc)m(he)315 1835 y(Grupp)s(e)39
b Fg(Q)p Fh(\()p Fg(M)10 b Fh(\))40 b(und)f(einen)h(Monoidhomomorphism)
m(us)i Fg(\023)d Fh(:)g Fg(M)49 b Fe(\000)-59 b(!)38
b Fg(Q)p Fh(\()p Fg(M)10 b Fh(\))40 b(so)f(gibt,)315
1952 y(da\031)32 b(zu)i(jeder)f(ab)s(elsc)m(hen)i(Grupp)s(e)e
Fg(A)g Fh(und)h(zu)f(jedem)h(Monoidhomomorphism)m(us)i
Fg(f)j Fh(:)315 2068 y Fg(M)53 b Fe(\000)-60 b(!)42 b
Fg(A)f Fh(genau)g(ein)h(Grupp)s(enhomomorphism)m(us)2377
2042 y(\026)2356 2068 y Fg(f)53 b Fh(:)42 b Fg(Q)p Fh(\()p
Fg(M)10 b Fh(\))43 b Fe(\000)-59 b(!)42 b Fg(A)f Fh(so)g(existiert,)315
2184 y(da\031)32 b(das)h(folgende)g(Diagramm)f(k)m(omm)m(utiert:)1514
2327 y Fg(M)220 b(Q)p Fh(\()p Fg(M)10 b Fh(\))p 1647
2304 152 4 v 1716 2302 a Fd(-)1710 2283 y Fc(\023)1920
2726 y Fg(A)1660 2514 y Fc(f)1635 2453 y Fd(@)1718 2536
y(@)1801 2619 y(@)1805 2624 y(@)-83 b(R)p 1955 2624 4
254 v 68 w(?)2010 2501 y Fb(\026)1995 2519 y Fc(f)315
2849 y Fh(\(Hin)m(w)m(eis:)34 b(Vgl.)f(Konstruktion)g(der)g(rationalen)
g(Zahlen)g(aus)g(den)g(ganzen)g(Zahlen.\))148 2965 y(\(2\))42
b(Zeigen)36 b(Sie,)g(da\031)g(es)g(zu)h(jedem)f(k)m(omm)m(utativ)m(en)j
(Monoid)d Fg(M)47 b Fh(eine)36 b(ab)s(elsc)m(he)i(Grupp)s(e)315
3082 y Fg(Q)p Fh(\()p Fg(M)10 b Fh(\))42 b(und)g(einen)g
(Monoidhomomorphism)m(us)i Fg(\023)f Fh(:)g Fg(M)53 b
Fe(\000)-59 b(!)42 b Fg(Q)p Fh(\()p Fg(M)10 b Fh(\))42
b(so)g(gibt,)f(da\031)g(zu)315 3198 y(jeder)25 b(ab)s(elsc)m(hen)j
(Grupp)s(e)d Fg(A)g Fh(und)h(zu)g(jedem)g(Monoidhomomorphism)m(us)i
Fg(f)38 b Fh(:)28 b Fg(M)38 b Fe(\000)-59 b(!)27 b Fg(A)315
3314 y Fh(genau)34 b(ein)g(Grupp)s(enhomomorphism)m(us)1935
3288 y(\026)1913 3314 y Fg(f)41 b Fh(:)30 b Fg(Q)p Fh(\()p
Fg(M)10 b Fh(\))31 b Fe(\000)-60 b(!)30 b Fg(A)j Fh(so)h(existiert,)i
(da\031)e Fg(f)40 b Fh(=)3528 3288 y(\026)3507 3314 y
Fg(f)11 b(\023)315 3430 y Fh(gilt.)0 3594 y Fj(Aufgab)s(e)46
b(22.)f Fh(Sei)40 b Fe(V)47 b Fh(:)38 b Fj(T)-9 b(op)39
b Fe(\000)-60 b(!)39 b Fj(Me)g Fh(der)h(V)-8 b(ergi\031funktor.)39
b(F)2449 3598 y(\177)2446 3594 y(ur)g Fg(X)46 b Fe(2)39
b Fj(Me)h Fh(sei)g(Dis\()p Fg(X)8 b Fh(\))38 b(:=)0 3711
y(\()p Fg(X)r(;)17 b Fe(P)8 b Fh(\()p Fg(X)g Fh(\)\))33
b(der)g(diskrete)h(top)s(ologisc)m(he)f(Raum)g(auf)f
Fg(X)8 b Fh(.)33 b(Zeigen)f(Sie:)0 3827 y(Ist)h(\()p
Fg(Y)5 b(;)17 b(T)d Fh(\))32 b(ein)h(b)s(eliebiger)h(top)s(ologisc)m
(her)f(Raum)g(und)g Fg(f)38 b Fh(:)28 b Fg(X)36 b Fe(\000)-60
b(!)28 b Fg(Y)53 b Fh(eine)34 b(Mengenabbildung,)0 3943
y(so)k(ist)g Fg(f)47 b Fh(:)36 b(Dis\()p Fg(X)8 b Fh(\))36
b Fe(\000)-59 b(!)35 b Fh(\()p Fg(Y)5 b(;)17 b(T)d Fh(\))37
b(stetig,)h(d.h.)h(es)f(gibt)g(genau)f(einen)i(Morphism)m(us)3158
3917 y(\026)3137 3943 y Fg(f)47 b Fh(:)36 b(Dis\()p Fg(X)8
b Fh(\))0 4059 y Fe(\000)-60 b(!)28 b Fh(\()p Fg(Y)5
b(;)17 b(T)d Fh(\))32 b(in)g Fj(T)-9 b(op)33 b Fh(mit)g
Fg(f)38 b Fh(=)28 b Fe(V)8 b Fh(\()1260 4033 y(\026)1239
4059 y Fg(f)i Fh(\))28 b(=)g Fe(V)8 b Fh(\()1595 4033
y(\026)1574 4059 y Fg(f)j Fh(\))17 b(id)1768 4074 y Fc(X)1836
4059 y Fh(.)0 4224 y Fj(Aufgab)s(e)39 b(23.)i Fh(Sei)33
b Fg(A)g Fh(eine)h(\(additiv)m(e\))f(ab)s(elsc)m(he)i(Grupp)s(e.)d(Sei)
541 4381 y(T)-8 b(or\()p Fg(A)p Fh(\))28 b(:=)f Fe(f)p
Fg(a)p Fe(j)p Fg(a)h Fe(2)g Fg(A)33 b Fh(und)g(es)g(gibt)g
Fg(n)28 b Fe(2)g Fa(N)p Fg(;)49 b(n)28 b Fe(6)p Fh(=)f(0)33
b(mit)g Fg(na)28 b Fh(=)f(0)p Fe(g)p Fg(;)0 4537 y Fh(die)37
b Ff(T)-7 b(orsionsunter)i(grupp)g(e)35 b Fh(v)m(on)i
Fg(A)p Fh(.)f(Die)g(Grupp)s(e)h Fg(A)f Fh(hei\031t)h
Ff(torsionsfr)-5 b(ei)p Fh(,)35 b(w)m(enn)j(T)-8 b(or\()p
Fg(A)p Fh(\))34 b(=)g(0.)0 4654 y(Sei)f(T)-8 b(orfr\()p
Fg(A)p Fh(\))28 b(:=)f Fg(A=)17 b Fh(T)-8 b(or\()p Fg(A)p
Fh(\))32 b(und)h Fg(\027)i Fh(:)27 b Fg(A)h Fe(\000)-59
b(!)27 b Fh(T)-8 b(orfr\()p Fg(A)p Fh(\))32 b(die)h(k)-5
b(anonisc)m(he)35 b(Pro)5 b(jektion.)0 4770 y(Zeigen)44
b(Sie:)h(T)-8 b(orfr\()p Fg(A)p Fh(\))44 b(ist)g(torsionsfrei,)h(und)g
(zu)f(jeder)h(torsionsfreien)g(ab)s(elsc)m(hen)h(Grupp)s(e)0
4886 y Fg(B)e Fh(und)39 b(jedem)h(Homomorphism)m(us)g
Fg(f)49 b Fh(:)38 b Fg(A)g Fe(\000)-59 b(!)37 b Fg(B)44
b Fh(gibt)39 b(es)g(genau)g(einen)h(Homomorphism)m(us)21
4976 y(\026)0 5002 y Fg(f)h Fh(:)30 b(T)-8 b(orfr\()p
Fg(A)p Fh(\))30 b Fe(\000)-59 b(!)29 b Fg(B)5 b Fh(,)35
b(so)f(da\031)g Fg(f)40 b Fh(=)1353 4976 y(\026)1331
5002 y Fg(f)11 b(\027)6 b Fh(.)35 b(W)-8 b(elc)m(hes)36
b(P)m(aar)e(adjungierter)h(F)-8 b(unktoren)34 b(wird)h(durc)m(h)0
5119 y(dieses)f(univ)m(erselle)i(Problem)d(b)s(esc)m(hrieb)s(en?)0
5283 y Fj(Aufgab)s(e)g(24.)39 b Fh(Sei)28 b Fg(V)50 b
Fe(2)28 b Fg(k)s Fh(-)o Fj(V)-9 b(ek)p Fh(.)28 b(Zeigen)h(Sie,)f
(da\031)g(der)h(F)-8 b(unktor)28 b(Hom\()p Fg(V)5 b(;)17
b Fh(-\))27 b(:)h Fg(k)s Fh(-)p Fj(V)-9 b(ek)27 b Fe(\000)-60
b(!)28 b Fj(Me)0 5399 y Fh(einen)34 b(linksadjungierten)g(F)-8
b(unktor)33 b(b)s(esitzt.)100 5563 y(Abgab)s(e:)f(Dienstag,)h(den)g
(13.06.2000)e(11:15)h(Uhr)h(in)g(der)g(V)-8 b(orlesung)p
eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF