{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Text Outp ut" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple P lot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" -1 14 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 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 61 "Abstract Algebra with Map le\nChapter 2: Introduction to Groups" }}{PARA 19 "" 0 "" {TEXT -1 3 " by " }{URLLINK 17 "Alec Mihailovs" 4 "http://webpages.shepherd.edu/ami hailo" "" }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 "2: In troduction to Groups" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 "Cyclic an d Dihedral Groups" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "In this sect ion we will study two series of groups, cyclic and dihedral, represent ed as rotations and symmetries of regular polygons. It is convenient t o enumerate all the vertices of a regular " }{TEXT 265 1 "n" }{TEXT -1 32 "-gon counterclockwise from 1 to " }{TEXT 266 1 "n" }{TEXT -1 39 ". Now, if a symmetry moves vertex 1 to " }{XPPEDIT 18 0 "a[1];" "6 #&%\"aG6#\"\"\"" }{TEXT -1 14 ", vertex 2 to " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 25 ", and so on, ..., vertex " }{TEXT 269 1 "n" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" } {TEXT -1 25 ", then we can denote it [" }{XPPEDIT 18 0 "a[1];" "6#&%\" aG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 32 "]. For example, rotation of 360/" }{TEXT 267 1 "n" }{TEXT -1 68 " degrees moves vertex 1 to 2, vertex 2 to 3, and so on, ..., verte x " }{TEXT 268 1 "n" }{TEXT -1 43 " to 1, so it can be denoted as [2, \+ 3, ..., " }{TEXT 270 1 "n" }{TEXT -1 48 ", 1], which can be represente d in Maple as [$2.." }{TEXT 271 1 "n" }{TEXT -1 221 ",1]. The identity (i.e. no change) will be represented as [1, 2, ..., n], or [$1..n] in Maple notation. See Maple help item on \"dollar\", explaining that no tation. Now we can define cyclic and dihedral groups as follows: " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "cyclic:=n->[seq([$i..n,$1.. i-1],i=1..n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "dihedral: =n->[op(cyclic(n)),seq([n+1-j$j=i..n,n+1-j$j=1..i-1],i=1..n)]:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Cyclic groups are defined for all \+ positive integers " }{TEXT 278 2 "n," }{TEXT -1 47 " but dihedral grou ps are defined here only for " }{TEXT 279 1 "n" }{TEXT -1 17 " not les s than 3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dihedral(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$\"\"\"\"\"#7$F&F%F'F$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Here are correctly defined groups: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "cyclic(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7%\"\"\"\"\"#\"\"$7%F&F'F%7%F'F%F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dihedral(4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7*7&\"\"\"\"\"#\"\"$\"\"%7&F&F'F(F%7&F'F(F%F&7&F(F %F&F'7&F(F'F&F%7&F'F&F%F(7&F&F%F(F'7&F%F(F'F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "In this notation, the last element of " }{XPPEDIT 18 0 "D[4];" "6#&%\"DG6#\"\"%" }{TEXT -1 457 " transfers vertex 1 to itse lf, vertex 2 to the position of vertex 4, vertex 3 to itself, and vert ex 4 to the position of vertex 2, so it is the reflection across the d iagonal connecting 1 and 3, see the picture below. Permutation notatio n introduced above, is rather long. To make notation easier, we will d enote elements just by their ordinal numbers in the lists of the group elements, so the identity will always be the number 1, and the last e lement of " }{XPPEDIT 18 0 "D[4];" "6#&%\"DG6#\"\"%" }{TEXT -1 13 " wi ll be 8. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "In this new notatio n, to find the inverse elements, we can use the following procedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "inv:=proc(a,g) local i,v ,b,k; k:=nops(g[a]); b:=[0$k]; for i to k do b[g[a][i]]:=i od; member( b,g,'v'); v end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "For example, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "inv(5,cyclic(12));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "inv(5,dihedral(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "If we want to see the \+ graphical representations of " }{TEXT 263 2 "kr" }{TEXT -1 13 " elemen ts of " }{TEXT 260 1 "g" }{TEXT -1 16 ", starting from " }{TEXT 262 1 "m" }{TEXT -1 13 ", displaying " }{TEXT 261 1 "k" }{TEXT -1 26 " eleme nts in each row, in " }{TEXT 282 1 "r" }{TEXT -1 42 " rows, we can use the following procedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "with(plots): with(plottools): " }}{PARA 7 "" 1 "" {TEXT -1 50 "War ning, the name changecoords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 890 "Grid:=proc(g,m,k,r) local ngon,ngo ns1,ngons2,n,a,b,i,j1,j2,p,ar,l,ngonlabels,text1,text2,textar;\nn:=nop s(g[1]); \nngon := (a,b) -> [seq([ a+cos(2*Pi*i/n), b+sin(2*Pi*i/n) ], i = 1..n)]:\nngons1:=seq(seq(ngon(10*j1,-4*j2),j1=1..k),j2=1..r);\nng ons2:=seq(seq(ngon(10*j1+4.5,-4*j2),j1=1..k),j2=1..r);\np:=polygonplot (\{ngons1,ngons2\},axes=NONE,scaling=CONSTRAINED,color=aquamarine):\na r:=seq(seq(arrow([10*j1+1.6,-4*j2],vector([1.2,0]),.05,.3,.3,color=blu e),j1=1..k),j2=1..r):\nngonlabels:=(a,b,l)->seq([ a+1.4*cos(2*Pi*i/n), b+1.4*sin(2*Pi*i/n),l[i+1] ], i = 0..n-1):\ntext1:=textplot([seq(seq( ngonlabels(10*j1,-4*j2,[$1..n]),j1=1..k),j2=1..r)],color=red): \ntext2 :=textplot([seq(seq(ngonlabels(10*j1+4.5,-4*j2,g[inv(m-1+j1+k*(j2-1),g )]),j1=1..k),j2=1..r)],color=red):\ntextar:=textplot([seq(seq([10*j1+2 .2,-4*j2+.5,m-1+j1+k*(j2-1)],j1=1..k),j2=1..r)],color=blue):\ndisplay( p,ar,text1,text2,textar) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 " Here is the list of elements of " }{XPPEDIT 18 0 "D[4];" "6#&%\"DG6#\" \"%" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Grid (dihedral(4),1,2,4);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6gp-%)POLYGONSG637&7$$\"$X#!\"\"$!\"(\"\"!7$$\"$N#F*$!\") F-7$F($!\"*F-7$$\"$b#F*F17&7$$\"$X\"F*$!#6F-7$$\"$N\"F*$!#7F-7$F;$!#8F -7$$\"$b\"F*FB7&7$F($!#:F-7$F/$!#;F-7$F($!#F-Fbo7$F\\pFeo7$$\"#@F-Fbo 7&7$F[oF+7$F`oF17$F[oF47$FhoF17&7$F\\pF+7$F_pF17$F\\pF47$FcpF17&7$F[oF =7$F`oFB7$F[oFE7$FhoFB7&7$F\\pF=7$F_pFB7$F\\pFE7$FcpFB7&7$F[oFL7$F`oFO 7$F[oFR7$FhoFO7&7$F\\pFL7$F_pFO7$F\\pFR7$FcpFO7&7$F;F]o7$F@Fbo7$F;Feo7 $FHFbo7&7$F(F]o7$F/Fbo7$F(Feo7$F7Fbo7&7$F;F+7$F@F17$F;F47$FHF1-%'COLOU RG6&%$RGBG$\")p:#R%F2$\")`B)e)F2$\")fqkdF2-F$6&7&7$$\"$;\"F*$!++++DSF5 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"Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36 " "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "C urve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55 " "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "C urve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74 " "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "C urve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93 " "Curve 94" "Curve 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "C urve 100" "Curve 101" "Curve 102" "Curve 103" "Curve 104" "Curve 105" "Curve 106" "Curve 107" "Curve 108" "Curve 109" "Curve 110" "Curve 111 " "Curve 112" "Curve 113" "Curve 114" "Curve 115" "Curve 116" "Curve 1 17" "Curve 118" "Curve 119" "Curve 120" "Curve 121" "Curve 122" "Curve 123" "Curve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Cur ve 129" "Curve 130" "Curve 131" "Curve 132" "Curve 133" "Curve 134" "C urve 135" "Curve 136" "Curve 137" "Curve 138" "Curve 139" "Curve 140" "Curve 141" "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146 " "Curve 147" "Curve 148" "Curve 149" "Curve 150" "Curve 151" "Curve 1 52" "Curve 153" "Curve 154" "Curve 155" "Curve 156" "Curve 157" "Curve 158" "Curve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Cur ve 164" "Curve 165" "Curve 166" "Curve 167" "Curve 168" "Curve 169" "C urve 170" "Curve 171" "Curve 172" "Curve 173" "Curve 174" "Curve 175" "Curve 176" "Curve 177" "Curve 178" "Curve 179" "Curve 180" "Curve 181 " "Curve 182" "Curve 183" "Curve 184" "Curve 185" "Curve 186" "Curve 1 87" "Curve 188" "Curve 189" "Curve 190" "Curve 191" "Curve 192" "Curve 193" "Curve 194" "Curve 195" "Curve 196" "Curve 197" "Curve 198" "Cur ve 199" "Curve 200" "Curve 201" "Curve 202" "Curve 203" "Curve 204" "C urve 205" "Curve 206" "Curve 207" "Curve 208" "Curve 209" "Curve 210" "Curve 211" "Curve 212" "Curve 213" "Curve 214" "Curve 215" "Curve 216 " "Curve 217" "Curve 218" "Curve 219" "Curve 220" "Curve 221" "Curve 2 22" "Curve 223" "Curve 224" "Curve 225" "Curve 226" "Curve 227" "Curve 228" "Curve 229" "Curve 230" "Curve 231" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Also, we can use this procedure to display only one eleme nt. For example, 11th element in " }{XPPEDIT 18 0 "D[20];" "6#&%\"DG6# \"#?" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Gri d(dihedral(20),11,1,1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6P-%)POLYGONSG6%767$$\"+_c5&4\"!\")$!+1I)4p$!\"*7$$\"+*p, 43\"F*$!+[Z@7MF-7$$\"+D&y(e5F*$!+1I)4>$F-7$$\"+*p,4.\"F*$!+%[V*[IF-7$$ \"#5\"\"!$!\"$F@7$$\"+1I)4p*F-F;7$$\"+[Z@7%*F-F67$$\"+1I)4>*F-F17$$\"+ %[V*[!*F-F+7$$\"\"*F@$!\"%F@7$FM$!+%*p,4VF-7$FJ$!+__y(e%F-7$FG$!+%*p,4 [F-7$FD$!+;l0^\\F-7$F>$!\"&F@7$F9Fhn7$F4Fen7$F/FX7$F(FU7$$\"#6F@FR767$ $\"+_c5X:F*F+7$$\"+*p,4`\"F*F17$$\"+D&y(3:F*F67$$\"+*p,4[\"F*F;7$$\"$X \"!\"\"FA7$$\"+,$)4>9F*F;7$$\"+v9A\"R\"F*F67$$\"+,$)4p8F*F17$$\"+[V*[N \"F*F+7$$\"$N\"FdpFR7$F_qFU7$F\\qFX7$FipFen7$FfpFhn7$FbpF[o7$F_pFhn7$F \\pFen7$FioFX7$FfoFU7$$\"$b\"FdpFR-%'COLOURG6&%$RGBG$\")p:#R%F*$\")`B) e)F*$\")fqkdF*-F$6&7&7$$\"$;\"Fdp$!++++DSF-7$F^s$!++++vRF-7$$\"++++W7F *Fcs7$FfsF`s7%7$Ffs$!++++]QF-7$$\"$G\"FdpFR7$Ffs$!++++]TF--%&STYLEG6#% ,PATCHNOGRIDG-Far6&Fcr$F@F@Fit$\"*++++\"F*-%'CURVESG6#7*F]sFbsFesFjsF] tF`tFhsF]s-%%TEXTG6%7$$\"$9\"FdpFRQ\"16\"-Far6&FcrFjtFitFit-Fau6%7$$\" +7z9L6F*$!+3iPnNF-Q\"2FguFhu-Fau6%7$$\"+zBE86F*$!+Z15xJF-Q\"3FguFhu-Fa u6%7$$\"+N**G#3\"F*$!+3iPnGF-Q\"4FguFhu-Fau6%7$$\"+zBEV5F*$!+x3_oEF-Q \"5FguFhu-Fau6%7$F>$!#EFdpQ\"6FguFhu-Fau6%7$$\"+4iPn&*F-FgwQ\"7FguFhu- Fau6%7$$\"+Z15x\"*F-F_wQ\"8FguFhu-Fau6%7$$\"+3iPn))F-FgvQ\"9FguFhu-Fau 6%7$$\"+x3_o')F-F_vQ#10FguFhu-Fau6%7$$\"#')FdpFRQ#11FguFhu-Fau6%7$Fey$ !+#zBEV%F-Q#12FguFhu-Fau6%7$F_y$!+`$**G#[F-Q#13FguFhu-Fau6%7$Fix$!+#zB E8&F-Q#14FguFhu-Fau6%7$Fcx$!+B\"z9L&F-Q#15FguFhu-Fau6%7$F>$!#aFdpQ#16F guFhu-Fau6%7$FewFc[lQ#17FguFhu-Fau6%7$F]wF][lQ#18FguFhu-Fau6%7$FevFgzQ #19FguFhu-Fau6%7$F]vFazQ#20FguFhu-Fau6%7$$\"$f\"FdpFRF]zFhu-Fau6%7$$\" +7z9$e\"F*F_vFczFhu-Fau6%7$$\"+zBEj:F*FgvFizFhu-Fau6%7$$\"+N**GK:F*F_w F_[lFhu-Fau6%7$$\"+zBE$\\\"F*FgwFe[lFhu-Fau6%7$FbpF]xF[\\lFhu-Fau6%7$$ \"+@wt19F*FgwF_\\lFhu-Fau6%7$$\"+l+rn8F*F_wFc\\lFhu-Fau6%7$$\"+@wtO8F* FgvFg\\lFhu-Fau6%7$$\"+)3_oJ\"F*F_vF[]lFhu-Fau6%7$$\"$J\"FdpFRFfuFhu-F au6%7$Fj_lFazFavFhu-Fau6%7$Fe_lFgzFivFhu-Fau6%7$F`_lF][lFawFhu-Fau6%7$ F[_lFc[lFiwFhu-Fau6%7$FbpFi[lF_xFhu-Fau6%7$Fc^lFc[lFexFhu-Fau6%7$F^^lF ][lF[yFhu-Fau6%7$Fi]lFgzFayFhu-Fau6%7$Fd]lFazFgyFhu-Fau6%7$$\"$A\"Fdp$ !#NFdpF]zFgt-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Cur ve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 2 2" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" " Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curv e 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41 " "Curve 42" "Curve 43" "Curve 44" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "If we want to see its representation as a permutation, it can b e done as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dihed ral(20)[11];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#76\"#6\"#7\"#8\"#9\"#: \"#;\"#<\"#=\"#>\"#?\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Cyclic group " }{XPPEDIT 18 0 "C[n];" "6#&%\"CG6#%\"nG" }{TEXT -1 5 " has " }{TEXT 264 2 "n " } {TEXT -1 28 "elements and dihedral group " }{XPPEDIT 18 0 "D[n];" "6#& %\"DG6#%\"nG" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "2*n;" "6#*&\"\"#\"\" \"%\"nGF%" }{TEXT -1 20 " elements. Using the" }{TEXT 283 5 " nops" } {TEXT -1 22 " command can test it: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "nops(cyclic(12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nops(dihedral(100)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$+#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Notice, that all elements of cyclic groups are rotations. The first " }{TEXT 256 1 "n" }{TEXT -1 13 " elements of " }{TEXT 257 8 "dihedral" }{TEXT -1 1 "(" }{TEXT 258 1 "n" }{TEXT -1 27 ") are rota tions, the other " }{TEXT 259 1 "n" }{TEXT -1 18 " are reflections. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "To multiply elements, we can us e the following procedure: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "mult:=proc(a,b,g) local i,v; member([seq(g[a][g[b][i]],i=1..nops(g [a]))],g,'v');v end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For examp le, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "mult(3,7,cyclic(12) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "mult(3,7,dihedral(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "mul t(7,3,dihedral(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "Cayley Tables" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 61 "The following procedure displays the Cayley table \+ of a group:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "cayley:=g->M atrix(nops(g),(i,j)->mult(i,j,g)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "For example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "cayl ey(dihedral(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")/F> 7-%'MATRIXG6#7*7*\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7*F-F.F/F,F3 F0F1F27*F.F/F,F-F2F3F0F17*F/F,F-F.F1F2F3F07*F0F1F2F3F,F-F.F/7*F1F2F3F0 F/F,F-F.7*F2F3F0F1F.F/F,F-7*F3F0F1F2F-F.F/F," }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 281 "It is easy to see that the product of two reflections \+ (i.e. numbers from 5 to 8) is a rotation, and the product of a reflect ion and a rotation is a rotation. Another evident thing is that reflec tions are inverse to themselves. Notice that the matrix is not symmetr ic because group " }{XPPEDIT 18 0 "D[4];" "6#&%\"DG6#\"\"%" }{TEXT -1 80 " is not Abelian. The following procedure is checking whether a gro up is Abelian:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "isAbelian :=g->IsMatrixShape(cayley(g),symmetric):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "isAbelian(dihedral(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "For cyclic grou ps Cayley tables are very symmetric:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "cayley(cyclic(12));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RTABLEG6+\")O(zB\"%)anythingG%'MatrixG%,rectangularG%.Fortran_ord erG7\"\"\"#;\"\"\"\"#7F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 289 "It i s a placeholder. By default, Maple shows them for matrices larger than 10x10. To work with the matrix, one should right-click on the placeho lder and use the context menu. In case we want to see the matrix in th e worksheet instead of that, we should increase the default size of rt able:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(rtablesi ze=25):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "%;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")O(zB\"-%'MATRIXG6#7.7.\"\"\"\"\"# \"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#77.F-F.F/F0F1F2F3F4F5F6F 7F,7.F.F/F0F1F2F3F4F5F6F7F,F-7.F/F0F1F2F3F4F5F6F7F,F-F.7.F0F1F2F3F4F5F 6F7F,F-F.F/7.F1F2F3F4F5F6F7F,F-F.F/F07.F2F3F4F5F6F7F,F-F.F/F0F17.F3F4F 5F6F7F,F-F.F/F0F1F27.F4F5F6F7F,F-F.F/F0F1F2F37.F5F6F7F,F-F.F/F0F1F2F3F 47.F6F7F,F-F.F/F0F1F2F3F4F57.F7F,F-F.F/F0F1F2F3F4F5F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Right-clicking still shows the same conte xt menu. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "isAbelian(cycl ic(12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Operations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "If we need to do a lot of calculations in some specific g roup, " }{XPPEDIT 18 0 "D[4];" "6#&%\"DG6#\"\"%" }{TEXT -1 92 ", for i nstance, we can define special multiplication and inverse element oper ations for it: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "`&*`:=(a ,b)->mult(a,b,Group):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Before using it, we should specify the group:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Group:=dihedral(4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "3&*4;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Similarly for the inverse element, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "`&-`:=a->inv(a,Group):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "&-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "2&*5&*&-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "One has to be very car eful with that though, since the answers depend on the group, and for \+ calculations in other groups we should redefine the " }{TEXT 284 5 "Gr oup" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Grou p:=cyclic(12):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "3&*4;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "2&*5&*&-2;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Exerc ises" }}{EXCHG {PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 272 1 "1" }{TEXT -1 19 ". Draw elements of " }{XPPEDIT 18 0 "D[3];" "6#&%\"DG6#\"\"$" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 2 ". " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 280 1 "2" }{TEXT -1 23 ". Draw 15th element of " }{XPPEDIT 18 0 "D[24];" "6#&%\"DG6#\"#C" } {TEXT -1 2 ". " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 281 1 "3" } {TEXT -1 28 ". Represent 15th element of " }{XPPEDIT 18 0 "D[24];" "6# &%\"DG6#\"#C" }{TEXT -1 18 " as a permutation." }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 273 1 "4" }{TEXT -1 27 ". Display Cayley tables o f " }{XPPEDIT 18 0 "D[3];" "6#&%\"DG6#\"\"$" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "C[6];" "6#&%\"CG6#\"\"'" }{TEXT -1 28 ". Are these grou ps Abelian? " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 274 1 "5" } {TEXT -1 7 ". Find " }{XPPEDIT 18 0 "a*b*a^`-1`;" "6#*(%\"aG\"\"\"%\"b GF%)F$%#-1GF%" }{TEXT -1 26 "for all pairs of elements " }{TEXT 275 1 "a" }{TEXT -1 5 " and " }{TEXT 276 1 "b" }{TEXT -1 6 " from " } {XPPEDIT 18 0 "D[3];" "6#&%\"DG6#\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "D[4];" "6#&%\"DG6#\"\"%" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 277 1 "6" }{TEXT -1 30 ". Guess if 11&*17&*27&*&- 3 in " }{XPPEDIT 18 0 "D[15];" "6#&%\"DG6#\"#:" }{TEXT -1 74 " is a ro tation, or a reflection, and check it out by direct calculation. " }} }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 12192704 12379736 }{RTABLE M7R0 I5RTABLE_SAVE/12192704X,%)anythingG6"6"[gl!"%!!!#[o")")"""""#""$""%""&""'""("") F(F)F*F'F,F-F.F+F)F*F'F(F-F.F+F,F*F'F(F)F.F+F,F-F+F.F-F,F'F*F)F(F,F+F.F-F(F'F*F )F-F,F+F.F)F(F'F*F.F-F,F+F*F)F(F'F& } {RTABLE M7R0 I5RTABLE_SAVE/12379736X,%)anythingG6"6"[gl!"%!!!#[t"-"-"""""#""$""%""&""'""("") ""*"#5"#6"#7F(F)F*F+F,F-F.F/F0F1F2F'F)F*F+F,F-F.F/F0F1F2F'F(F*F+F,F-F.F/F0F1F2F 'F(F)F+F,F-F.F/F0F1F2F'F(F)F*F,F-F.F/F0F1F2F'F(F)F*F+F-F.F/F0F1F2F'F(F)F*F+F,F. F/F0F1F2F'F(F)F*F+F,F-F/F0F1F2F'F(F)F*F+F,F-F.F0F1F2F'F(F)F*F+F,F-F.F/F1F2F'F(F )F*F+F,F-F.F/F0F2F'F(F)F*F+F,F-F.F/F0F1F& }