{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(geom3d):with(pl ots):" }}{PARA 7 "" 1 "" {TEXT -1 56 "Warning, the names center and po lar have been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "cube(c,point(o,0,0,0),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "plotwuerfel:=display(textp lot3d(\{[2,2,2,'1'],[-2,2,2,'2'],[-2,-2,2,'3'],[2,-2,2,'4'],[2,2,-2,'5 '],[-2,2,-2,'6'],[-2,-2,-2,'7'],[2,-2,-2,'8']\},color=black),draw(c,ax es=boxed,style=line,thickness=5),orientation=[22,68]):\ndisplay(plotwu erfel);" }}{PARA 13 "" 1 "" {GLPLOT3D 562 426 426 {PLOTDATA 3 "6/-%%TE XTG6%7%$\"\"#\"\"!F'F'%\"1G-%'COLOURG6&%$RGBGF)F)F)-F$6%7%$!\"#F)F'F'% \"2GF+-F$6%7%F2F2F'%\"3GF+-F$6%7%F'F2F'%\"4GF+-F$6%7%F'F'F2%\"5GF+-F$6 %7%F2F'F2%\"6GF+-F$6%7%F2F2F2%\"7GF+-F$6%7%F'F2F2%\"8GF+-%)POLYGONSG6* 7&7%$\"+330K " 0 "" {MPLTEXT 1 0 45 "S:=\{[[1,2,3,4],[5,6,7,8]],[[2,4,5],[3,8,6]] \};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG<$7$7&\"\"\"\"\"#\"\"$\" \"%7&\"\"&\"\"'\"\"(\"\")7$7%F)F+F-7%F*F0F." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 13 "erzeugt wird." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(group);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7>%)DerivedSG%$L CSG%.NormalClosureG%,RandElementG%-SnConjugatesG%&SylowG%-areconjugate G%'centerG%,centralizerG%%coreG%'cosetsG%'cosrepG%(derivedG%)elementsG %,groupmemberG%+grouporderG%&interG%(invpermG%*isabelianG%)isnormalG%+ issubgroupG%)mulpermsG%+normalizerG%&orbitG%'parityG%(permrepG%%presG% +transgroupG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Vorsicht: Maple u nd GAP multiplizieren Permutationen von links, Lehrb\374cher meist von rechts:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "mulperms([[1,2] ,[3,4]],[[1,2,3,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$\"\"\"\" \"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "Folgende Funktion multipl iziert die Permutationen in Zykelschreibweise in der Liste L (die auch mehr als 2 Elemente haben kann) auf die von uns verwendete Weise (von rechts):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "mperm:=proc(L )\nlocal j,S,P;\nP:=L[1];\nfor j from 2 to nops(L) do\n P:=mulperms(L [j],P)\nod:\nreturn(P);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "mperm([ [[1,2],[3,4]] , [[1,2,3,4]] ]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7#7$\"\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "mperm([[[1,3]],[[1,2]],[[1,2,3,4]]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7&\"\"\"\"\"$\"\"%\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Folgende Funktion berechnet das Bild von x unter g f \374r die Operation von S_n auf \{1,...,n\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "applyperm:=proc(g,x)\nlocal j,jj;\nfor j from nop s(g) to 1 by -1 do\n for jj from 1 to nops(g[j]) do\n if g[j][jj]= x then\n if jj=nops(g[j]) then \n return(g[j][1]);\n \+ else\n return(g[j][jj+1]); \n fi;\n fi;\n od:\nod:\nr eturn(x);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "app lyperm([[1,2,3,4],[5,6,7,8]],8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Folgende Funktion berechn et die Bahn von x unter der Gruppe G erzeugt von den Permutationen in \+ der Menge S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "Orbit:=proc (S,x)\nlocal Gx,st,Gx1,j,jj;\nGx:=\{x\};\nst:=false;\nwhile st=false d o\n Gx1:=Gx;\n for j from 1 to nops(Gx) do\n for jj from 1 to nop s(S) do\n Gx1:=Gx1 union \{applyperm(S[jj],Gx[j])\};\n od:\n \+ od: \n if nops(Gx1)=nops(Gx) then st:=true;fi;\n Gx:=Gx1;if ver>0 th en print(Gx);fi;\nod:\nreturn(Gx);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ver:=1;#Ausgabe der Zwischenschritte;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$verG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Orbit(S,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\" \"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&\"\"\"\"\"#\"\"$\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<)\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"' \"\"(\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*\"\"\"\"\"#\"\"$\"\"% \"\"&\"\"'\"\"(\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*\"\"\"\"\"# \"\"$\"\"%\"\"&\"\"'\"\"(\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Folgende Funktion berechnet alle Elemente der Gruppe G erzeugt von S " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 323 "Elements:=proc(S)\nloc al G,st,G1,j,jj;\nG:=\{[]\};\nst:=false;\nwhile st=false do\n G1:=G; \n for j from 1 to nops(G) do\n for jj from 1 to nops(S) do\n \+ G1:=G1 union \{mperm([S[jj],G[j]])\};\n od:\n od: \n if nops(G1) =nops(G) then st:=true;fi;\n if ver>0 then print(G1 minus G,nops(G1)) ;fi;\n G:=G1;\nod:\nreturn(G);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "ver:=1;#Ausgabe der Zwischenschritte;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$verG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Elements(\{[[1,2,3]],[[2,3,4]]\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$<$7#7%\"\"\"\"\"#\"\"$7#7%F'F(\"\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<&7$7$\"\"\"\"\"#7$\"\"$\"\"%7#7%F&F)F'7$7$F&F)7$F 'F*7#7%F'F*F)\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<&7#7%\"\"\"\"\" #\"\"%7#7%F&\"\"$F(7#7%F&F(F'7#7%F&F(F+\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#7$7$\"\"\"\"\"%7$\"\"#\"\"$\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<\"\"#7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<.7\"7#7%\" \"\"\"\"#\"\"%7$7$F'F(7$\"\"$F)7#7%F'F(F-7#7%F(F-F)7#7%F'F-F(7$7$F'F-7 $F(F)7#7%F(F)F-7#7%F'F-F)7#7%F'F)F(7#7%F'F)F-7$7$F'F)7$F(F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ver:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$verG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$7$7&\"\"\"\"\"#\"\"$\"\"%7& \"\"&\"\"'\"\"(\"\")7$7%F'F)F+7%F(F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Elements(S);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<:7\"7$7&\"\"\"\"\"%\"\"$\"\"#7&\"\"&\"\")\"\"(\"\"'7$7%F'F)F/7% F(F.F,7$7%F'F/F)7%F(F,F.7$7&F'F,F-F(7&F*F/F.F)7$7&F'F*F/F,7&F)F.F-F(7& 7$F'F,7$F*F-7$F)F.7$F(F/7$7%F*F,F(7%F)F/F-7&7$F'F/7$F*F,7$F)F-7$F(F.7$ 7%F'F-F)7%F*F,F.7&7$F'F.F>7$F)F(7$F,F/7&7$F'F*7$F)F,F@7$F.F-7&7$F'F(F> FR7$F/F.7$7&F'F,F/F*7&F)F(F-F.7&FM7$F*F)F@7$F,F-7&7$F'F-7$F*F.7$F)F/7$ F(F,7$7&F'F(F-F,7&F*F)F.F/7$7%F'F-F/7%F*F(F.7&7$F'F)7$F*F(7$F,F.7$F/F- 7$7%F'F/F-7%F*F.F(7$7%F'F)F-7%F*F.F,7&FM7$F*F/FR7$F(F-7$7&F'F*F)F(7&F, F/F.F-7$7%F*F(F,7%F)F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Folgende Funktion berechnet den St abilisator von x unter Gruppe G erzeugt von S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "Stab:=proc(S,x)\nlocal j,St,G;\nSt:=\{\};\nG:=E lements(S);\nfor j from 1 to nops(G) do\n if applyperm(G[j],x)=x then St:=\{op(St),G[j]\};fi;\nod:\nreturn(St);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Stab(S,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%7\"7$7%\"\"#\"\"&\"\"%7%\"\"$\"\"'\"\")7$7%F'F)F(7%F+ F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(plotwuerfel );" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6/-%%TEXTG6% 7%$\"\"#\"\"!F'F'%\"1G-%'COLOURG6&%$RGBGF)F)F)-F$6%7%$!\"#F)F'F'%\"2GF +-F$6%7%F2F2F'%\"3GF+-F$6%7%F'F2F'%\"4GF+-F$6%7%F'F'F2%\"5GF+-F$6%7%F2 F'F2%\"6GF+-F$6%7%F2F2F2%\"7GF+-F$6%7%F'F2F2%\"8GF+-%)POLYGONSG6*7&7%$ \"+330K " 0 "" {MPLTEXT 1 0 491 "Orbitg:=proc(S,x)\nlocal Gx,st,Gx1,j,jj,g,newel; \nGx:=\{x\};\ng:=table();\ng[x]:=[];\nst:=false;\nwhile st=false do\n \+ Gx1:=Gx;\n for j from 1 to nops(Gx) do\n for jj from 1 to nops(S) do\n newel:=applyperm(S[jj],Gx[j]);\n if member(newel,Gx1)= false then\n Gx1:=Gx1 union \{newel\};#print(g);print(g[Gx[j]]) ;\n g[newel]:=mperm([S[jj],g[Gx[j]]]);\n fi;\n od:\n o d: \n if nops(Gx1)=nops(Gx) then st:=true;fi;\n Gx:=Gx1;if ver>0 the n print(Gx);fi;\nod:\nreturn([Gx,g]);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "print(Orbitg(S,1)[1]);\nprint(Orbitg(S,1)[2]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"' \"\"(\"\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7*/\"\"\"7\"/ \"\"#7$7&F(F+\"\"$\"\"%7&\"\"&\"\"'\"\"(\"\")/F.7&7$F(F.7$F+F/7$F1F37$ F2F4/F/7&7$F(F/7$F+F47$F.F17$F2F3/F17$7&F(F1F4F/7&F+F2F3F./F27$7%F(F2F 47%F+F3F//F37&7$F(F3F>7$F.F/7$F1F2/F47$7%F(F4F.7%F+F1F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Berechnung von Erzeugern von Stab_G(1) f \374r G erzeugt von S=\{h,m\} mit" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "h:=[[1, 2, 3, 4], [5, 6, 7, 8]];\nm:=[[2, 4, 5], [3, \+ 8, 6]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG7$7&\"\"\"\"\"#\"\"$ \"\"%7&\"\"&\"\"'\"\"(\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG7 $7%\"\"#\"\"%\"\"&7%\"\"$\"\")\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "mperm([invperm(mperm([m,h])),h,h,h]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%\"\"#\"\"%\"\"&7%\"\"$\"\")\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "mperm([h,m,h]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$7%\"\"#\"\"&\"\"%7%\"\"$\"\"'\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Folgende Funktion berechnet Erzeuger von \+ Stab(x) f\374r die Operation von G erzeugt von S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 404 "Stabgens:=proc(S,x)\nlocal j,Sg,Gxg,Gx,g,h,y ,hy,jj;\nSg:=\{\};\nGxg:=Orbitg(S,x);\nGx:=Gxg[1];g:=Gxg[2];\nfor jj f rom 1 to nops(S) do\n h:=S[jj];if ver>0 then print(h);fi;\n for j fr om 1 to nops(Gx) do\n y:=Gx[j];\n hy:=applyperm(h,y);\n if ver>0 then print(y,hy,g[y],g[hy],mperm([invperm(g[hy]),h,g[y]]));fi; \n Sg:=\{op(Sg),mperm([invperm(g[hy]),h,g[y]])\};\n od:\nod:\nret urn(Sg minus \{[]\});\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Stabgens(S,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$7 $7%\"\"#\"\"&\"\"%7%\"\"$\"\"'\"\")7$7%F&F(F'7%F*F,F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Somit haben wir mit der Bahnenformel f \374r die Anzal der Elemente von G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "8*3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Betrachte nun ein gr\366\337eres B eispiel:\nG sei die Gruppe erzeugt von S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "S:=\{[[1,2,3],[4,6,7],[5,8,9],[10,12,11]],[[1,3],[2,4 ],[7,8]]\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG<$7&7%\"\"\"\"\"# \"\"$7%\"\"%\"\"'\"\"(7%\"\"&\"\")\"\"*7%\"#5\"#7\"#67%7$F(F*7$F)F,7$F .F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Direkte Berechnung der Ele mente von G dauert 9 sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Elements(S):nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%'H\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Durch einmalige Anwendung der Bahn enformel:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Berechnung der Bahn \+ von 1 unter G dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "og:=Orbitg(S,1):G_1:=og[1];nops(G_1);gG_1:=og[2]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$G_1G<+\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\") \"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Direkte Berechnung von Stab(1) dauert ebenfalls 9 s ec, da wir alle Elemente von G aufz\344hlen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Stab(S,1):nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$W\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Bahnenformel \+ liefert" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "9*144;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%'H\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Berechnen von Erzeugern von Stab(1) dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gensStab1:=Stabgens(S,1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*gensStab1G<'7%7$\"\"#\"\")7$\"\"%\"\"(7$ \"\"&\"\"'7$7*F(F/F,\"\"*F.F)F+\"\"$7%\"#5\"#7\"#67$7*F(F3F+F)F.F2F,F/ 7%F5F7F67%7$F(F.7$F3F,7$F+F27%7$F(F+7$F3F27$F.F," }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 81 "Nun k\366nnen wir die Anahl der Elemente von Stab( 1) viel schneller berechnen (2sec)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "nops(Elements(gensStab1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$W\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Iteriere nun die Bahnenformel:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Berechn ung der Bahn von 2 unter Stab(1) dauert <<1sec" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "og:=Orbitg(gensStab1,2):Stab1_2:=og[1];nops(St ab1_2);gStab1_2:=og[2]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Stab1_2G <*\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Direkte B erechnung von Stab(1,2) dauert 2 sec, da wir alle Elemente von Stab(1, 2) aufz\344hlen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Stab(ge nsStab1,2);nops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<47\"7&7$\"\"$ \"\")7$\"\"%\"\"(7$\"\"'\"\"*7%\"#5\"#7\"#67&7$F'F*7$F-F(7$F+F.F/7%7%F 'F-F+7%F*F.F(F/7%F8F97%F0F2F17%7%F'F+F-7%F*F(F.F;7%F&F)F,7%F4F5F67&F&F )F,F;7#F/7%7$F'F.7$F*F-7$F+F(7$F8F97&FDFEFFF;7$F=F>7#F;7&F4F5F6F;7%F=F >F/7&FDFEFFF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Bahnenformel liefert" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "9*8*18;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%'H\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Berechn ung von Erzeugern von Stab(1,2) dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "gensStab12:=Stabgens(gensStab1,2);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%+gensStab12G<17&7$\"\"$\"\")7$\"\"%\"\"(7$\"\" '\"\"*7%\"#5\"#7\"#67&7$F(F+7$F.F)7$F,F/F07%7%F(F.F,7%F+F/F)F07%7%F(F, F.7%F+F)F/7%F1F3F27%F'F*F-7%F5F6F77&F'F*F-F>7#F07%7$F(F/7$F+F.7$F,F)7$ F9F:7&FDFEFFF>7$F7&F5F6F7F>7&FDFEFFF0" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 55 "Berechnung der Bahn von 3 unter Stab(1,2) dauert <<1sec " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "og:=Orbitg(gensStab12,3 ):Stab12_3:=og[1];nops(Stab12_3);gStab12_3:=og[2]:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)Stab12_3G<(\"\"$\"\"%\"\"'\"\"(\"\")\"\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Berechnung von Erzeugern von Stab(1,2) und daraus aller E lemente dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ge nsStab123:=Stabgens(gensStab12,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%,gensStab123G<$7#7%\"#5\"#7\"#67#7%F(F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Elements(gensStab123);nops(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<%7\"7#7%\"#5\"#7\"#67#7%F'F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "Iterativ e Anwendung der Bahnenformel liefert somit\n| G | = | G 1 | * | Stab(1 ) | \n = | G 1 | * | Stab(1) 2 | * |Stab(1,2)| \n = | G 1 \+ | * | Stab(1) 2 | * | Stab(1,2) 3 | * |Stab(1,2,3)|" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "9*8*6*3;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"%'H\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 253 "Insgesamt dauert di e Berechnung der Gruppenordnung von G durch\n- Aufz\344hlung aller 129 6 Elemente von G etwa 9 sec\n- einmalige Anwendung der Bahnenformel un d dem Satz zu Angabe von Erzeugern des Stabilisators etwa 2 sec\n- die iterative Anwendung << 1 sec\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "#Stab(gensStab12,3);nops(%); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "gensStab123:=Stabgens(g ensStab12,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "og:=Orbitg (gensStab123,4):Stab123_4:=og[1];gStab123_4:=og[2]:print(gStab123_4); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gensStab1234:=Stabgens( gensStab123,4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 219 "S:=\{[[1,2,3],[4,6,7],[5,8,9],[10,12,11]],[[2,4],[ 3,5],[10,6],[7,8],[9,11]]\};\n#gensG:=\{[[1,2,3],[4,6,7],[5,8,9],[10,1 2,11]],[[2,4],[3,5,10,6],[7,8],[9,11]]\};\n#gensG:=\{[[1,2,3],[4,6,7], [5,8,9],[10,12,11]],[[2,4],[7,8]]\};\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Betrachte nun ein gr\366\337eres Beispiel:\nG sei die Gru ppe erzeugt von S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "S:=\{ [[1,2,3],[4,6,7],[5,8,9],[10,12,11]],[[2,4],[3,5],[10,6],[7,8],[9,11]] \};\nS:=[[[2,3,4,5,6],[7,8,9,10,11]],[[1,4,10,11,6],[2,3,9,12,7]]];" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Direkte Berechnung der Elemente \+ dauert 33 sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Elements(S ):nops(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Durch einmalige Anw endung der Bahnenformel:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Berec hnung der Bahn von 1 unter G dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "og:=Orbitg(S,1):G_1:=og[1];nops(G_1);gG_1:=og[2]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Direkte Berechnung von Stab(1) dauert ebenfalls 33 sec, da wir alle Elemente von G aufz\344hlen." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Stab(S,1):nops(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Bahnenformel liefert" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "12*5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Berechnen von Erzeugern von Stab(1) dauert <<1sec" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gensStab1:=Stabgens(S,1); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Berechnung der Bahn von 2 unt er Stab(1) dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "og:=Orbitg(gensStab1,2):Stab1_2:=og[1];nops(Stab1_2);gStab1_2:=og[2]: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Direkte Berechnung von Stab(1 ,2) dauert 4 sec, da wir alle Elemente von Stab(1,2) aufz\344hlen:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Stab(gensStab1,2);nops(%); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Bahnenformel liefert" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "12*4*2;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 67 "Betrachte nun ein gr\366\337eres Beispiel:\nG sei \+ die Gruppe erzeugt von S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S:=[[[2,3,5,4]],[[1,5,3,6,2,4]]];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Direkte Berechnung der Elemente dauert 33 sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Elements(S):nops(%);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 43 "Durch einmalige Anwendung der Bahnenformel:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Berechnung der Bahn von 1 unter G \+ dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "og:=Orbitg (S,1):G_1:=og[1];nops(G_1);gG_1:=og[2]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Direkte Berechnung von Stab(1) dauert ebenfalls 33 sec, \+ da wir alle Elemente von G aufz\344hlen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Stab(S,1);nops(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Bahnenformel liefert" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "6*8;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Berechnen von Erze ugern von Stab(1) dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gensStab1:=Stabgens(S,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Berechnung de r Bahn von 2 unter Stab(1) dauert <<1sec" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "og:=Orbitg(gensStab1,2):Stab1_2:=og[1];nops(Stab1_2); gStab1_2:=og[2]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Direkte Berec hnung von Stab(1,2) dauert 4 sec, da wir alle Elemente von Stab(1,2) a ufz\344hlen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Stab(gensSt ab1,2);nops(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Bahnenform el liefert" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "6*4*2;" }}}} {MARK "78 0 0" 16 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }