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Cyclic \+ Groups" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "Primitive Roots" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Group " }{XPPEDIT 18 0 "Z[n]" "6#&% \"ZG6#%\"nG" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "phi(n);" "6#-%$phiG6# %\"nG" }{TEXT -1 48 " generators. To find them, we can use procedure \+ " }{TEXT 264 2 "un" }{TEXT -1 14 ". For example," }{TEXT 265 2 " " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "un:=n->select(m->evalb(igcd (m,n)=1),[$1..n]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "un(20) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"\"\"\"$\"\"(\"\"*\"#6\"#8\" #<\"#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The generators of " } {TEXT 256 1 "U" }{TEXT -1 1 "(" }{TEXT 259 1 "n" }{TEXT -1 28 ") if th ey exist, are called " }{TEXT 257 15 "primitive roots" }{TEXT -1 5 " m od " }{TEXT 258 1 "n" }{TEXT -1 46 ". One of them can be found using t he function " }{TEXT 260 8 "primroot" }{TEXT -1 10 " from the " } {TEXT 261 9 "numtheory" }{TEXT -1 60 " package that we already loaded \+ in section 2. For example, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "p rimroot(43);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "It fails when the group " }{TEXT 262 1 "U " }{TEXT -1 1 "(" }{TEXT 263 1 "n" }{TEXT -1 62 ") is not cyclic, so i t doesn't have a generator. For example, " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "primroot(45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%% FAILG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "To find all generators, \+ we can use the following procedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "primroots:=n->\{seq(primroot(n)^un(phi(n))[i] mod n, \+ i=1..phi(phi(n)))\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For examp le, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "primroots(43);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<.\"\"$\"\"&\"#7\"#=\"#>\"#?\"#E\"#G\" #H\"#I\"#L\"#M" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "primroots (45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#%%FAILG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The number of generators of " }{TEXT 266 1 "U" } {TEXT -1 1 "(" }{TEXT 267 1 "n" }{TEXT -1 9 ") equals " }{XPPEDIT 18 0 "phi(phi(n));" "6#-%$phiG6#-F$6#%\"nG" }{TEXT -1 41 " when it is a c yclic group. For example, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "phi(phi(43));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Elements of Order " }{TEXT 268 1 "d" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Theorem 4.4 on p. 80 of Dr. Gallian's text tells us that if " }{TEXT 294 1 "d" }{TEXT -1 26 " is a positive divisor of " }{TEXT 295 1 "n" }{TEXT -1 17 ", th en there are " }{XPPEDIT 18 0 "phi(d);" "6#-%$phiG6#%\"dG" }{TEXT -1 19 " elements of order " }{TEXT 296 1 "d" }{TEXT -1 4 " in " } {XPPEDIT 18 0 "Z[n];" "6#&%\"ZG6#%\"nG" }{TEXT -1 44 ". The following \+ procedure lists all of them:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "nordlist:=(d,n)->`if`(type(n/d,integer),map(x->x*n/d mod n,un(d) ),[]): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "For example, the list \+ of elements of order 20 in " }{XPPEDIT 18 0 "Z[100];" "6#&%\"ZG6#\"$+ \"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "nordl ist(20,100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"&\"#:\"#N\"#X\"# b\"#l\"#&)\"#&*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The following \+ procedure counts the number of elements of order " }{TEXT 271 1 "d" } {TEXT -1 4 " in " }{TEXT 272 1 "U" }{TEXT -1 1 "(" }{TEXT 273 1 "n" } {TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "nordU:=p roc(d,n) local i, k;\nk:=0; for i from 1 to phi(n) do if order(un(n)[i ],n)=d then k:=k+1 fi od; k end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "For example, the number of elements of order 2 in " }{TEXT 297 1 " U" }{TEXT -1 5 "(45):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "no rdU(2,45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "It immediately implies that " }{TEXT 274 1 "U" }{TEXT -1 164 "(45) is not a cyclic group, because a cyclic group migh t have either 0 elements of order 2 if it has an odd order, or 1 eleme nt of order 2 if it has an even order. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The following procedure counts the number of elements of \+ order " }{TEXT 269 1 "d" }{TEXT -1 22 " in an extended group " }{TEXT 270 1 "G" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "Nord:=proc(d,g) local i, k; \nk:=0; for i from 1 to nops(g[1]) do if Ord(g[1][i],g)=d then k:=k+1 fi od; k end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "For example, the number of elements of order 10 in \+ " }{TEXT 277 2 "SL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[5];" "6#&%\"Z G6#\"\"&" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Nord(10,SL2(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Notice that " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "phi(10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "and 24 is divisible by " } {XPPEDIT 18 0 "phi(10) = 4;" "6#/-%$phiG6#\"#5\"\"%" }{TEXT -1 85 ", a s it is supposed to be according to the Corollary on p. 80 of Dr. Gall ian's book. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Another example, \+ the number of elements of order 2 in groups " }{TEXT 275 2 "GL" } {TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[n];" "6#&%\"ZG6#%\"nG" }{TEXT -1 6 ") for " }{TEXT 276 1 "n" }{TEXT -1 14 " from 2 to 6: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "for n from 2 to 6 do Nord(2,GL2(n)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#b " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Try to find this sequence in \+ " }{URLLINK 17 "Neil Sloane's On-Line Encyclopedia of Integer Sequence s" 4 "http://akpublic.research.att.com/~njas/sequences/" "" }{TEXT -1 166 ". Certainly, for every specific group, one can write a program ca lculating the number of elements of given order faster. For instance, \+ for the elements of order 2 in " }{TEXT 278 2 "GL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[n];" "6#&%\"ZG6#%\"nG" }{TEXT -1 3 "), " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "ord2inGL2:=proc(n) local a,b,c,d,N ; N:=0;\nfor a from 0 to n-1 do for b from 0 to n-1 do for c from 0 to n-1 do for d from 0 to n-1 do \nif a^2+b*c mod n = 1 and b*(a+d) mod \+ n = 0 and c*(a+d) mod n = 0 and d^2+b*c mod n =1\nthen N:=N+1 fi od od od od; N-1 end: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The followin g procedure lists the elements of order " }{TEXT 279 1 "d" }{TEXT -1 22 " in an extended group " }{TEXT 280 1 "G" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "Nordlist:=proc(d,g) local i , v; v:=[];\nfor i from 1 to nops(g[1]) do if Ord(g[1][i],g)=d then v: =[op(v),g[1][i]] fi od; v end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "For example, here is the list of elements of order 2 in " }{TEXT 281 2 "GL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[5];" "6#&%\"ZG6#\"\"&" } {TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "map(matri x,Nordlist(2,GL2(5)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7A-%'matrixG 6#7$7$\"\"\"\"\"!7$F*\"\"%-F%6#7$F(7$F)F,-F%6#7$F(7$\"\"#F,-F%6#7$F(7$ \"\"$F,-F%6#7$F(7$F,F,-F%6#7$7$F,F*7$F*F)-F%6#7$FB7$F)F)-F%6#7$FB7$F5F )-F%6#7$FB7$F:F)-F%6#7$FB7$F,F)-F%6#7$FBF+-F%6#7$FCF(-F%6#7$FGF+-F%6#7 $FK7$F5F:-F%6#7$FO7$F5F5-F%6#7$FSFC-F%6#7$7$F*F57$F:F*-F%6#7$7$F)F5F+- F%6#7$F^o7$F)F:-F%6#7$7$F:F5Fjo-F%6#7$7$F,F5FC-F%6#7$7$F*F:7$F5F*-F%6# 7$F^pF+-F%6#7$Fjn7$F,F:-F%6#7$7$F:F:Ffp-F%6#7$FbqFC-F%6#7$F+FB-F%6#7$F 0F+-F%6#7$F4Ffq-F%6#7$F9Fbp-F%6#7$F>FC" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Subgroup Lattice of " }{XPPEDIT 18 0 "Z[n];" "6#&%\"ZG6#% \"nG" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "We have to \+ load two packages for this section:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "with(plottools): with(networks):" }}{PARA 7 "" 1 "" {TEXT -1 93 "Warning, these names have been redefined: dodecahedron, i cosahedron, octahedron, tetrahedron\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "A subgroup lattice of " }{XPPEDIT 18 0 "Z[n];" "6#&%\"ZG6 #%\"nG" }{TEXT -1 44 " can be drawn using the following procedure:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 524 "subZ:=proc(n) local d,i,j, k,x,G,len,L,LL;\nnew(G); d:=divisors(n);\naddvertex(map(x->cat(`<`,x m od n,`>`),d),G); len:=a->add(ifactors(a)[2,i,2],i=1..nops(ifactors(a)[ 2]));\nL:=seq(convert(select(a->evalb(len(a)=i),d),list),i=0..len(n)); \nfor i from 1 to len(n) do for j from 1 to nops(L[i]) do for k from 1 to nops(L[i+1]) do \nif L[i+1,k] mod L[i,j] = 0 then addedge(map(x->c at(`<`,x mod n,`>`),[L[i,j],L[i+1,k]]),G) fi od od od;\nLL:=seq(map(x- >cat(`<`,x mod n,`>`),L[i]),i=1..len(n)+1); \nrotate(draw(Linear(LL),G ),-Pi/2);\nend: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "For example, \+ for " }{TEXT 282 1 "n" }{TEXT -1 6 " = 30:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "subZ(30);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6?-%'CURVESG6$7$7$$\"\"#\"\"!$!\"#F*7$F($!\"$F*-%'COLOURG 6&%$RGBGF*$\"#5!\"\"F*-F$6$7$7$$F*F*F.7$$\"\"\"F*$!\"%F*F0-F$6$7$7$F=F +F-F0-F$6$7$FDF:F0-F$6$7$7$F=F.F6\"-F_o6$7$F=$!#HF6Q%<10>Feo-%'POINTSG6#F--F_ o6$7$F(FioQ%<15>Feo-F]p6#F<-F]p6#Ffn-F_o6$7$F=$!\"*F6Q$<1>Feo-F]p6#FU- F_o6$7$F;$!#>F6Q$<2>Feo-F]p6#FD-F_o6$7$F=FbqQ$<3>Feo-F]p6#F'-F_o6$7$F( FbqQ$<5>Feo-F]p6#F:-F_o6$7$F;FioQ$<6>Feo-F]p6#FK-%*AXESSTYLEG6#%%NONEG " 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve \+ 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Cur ve 28" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "It looks like a cube, d oesn't it? Even more after rotating the picture by 90 degrees:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rotate(%,Pi/2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6?-%'CURVESG6$7$7$$\"\"#\" \"!F(7$$\"\"$F*F(-%'COLOURG6&%$RGBGF*$\"#5!\"\"F*-F$6$7$7$F,$F*F*7$$\" \"%F*$\"\"\"F*F.-F$6$7$7$F(F=F+F.-F$6$7$FBF8F.-F$6$7$7$F,F=F:F.-F$6$7$ F'FIF.-F$6$7$F+F:F.-F$6$7$7$F(F9FIF.-F$6$7$FSF8F.-F$6$7$7$F=F=FSF.-F$6 $7$FZF'F.-F$6$7$FZFBF.-%%TEXTG6$7$$\"$3%!\"#F=Q$<0>6\"-F\\o6$7$$\"#HF4 F=Q%<10>Fco-%'POINTSG6#F+-F\\o6$7$FgoF(Q%<15>Fco-F[p6#F:-F[p6#FZ-F\\o6 $7$$\"\"*F4F=Q$<1>Fco-F[p6#FS-F\\o6$7$$\"#>F4F9Q$<2>Fco-F[p6#FB-F\\o6$ 7$F`qF=Q$<3>Fco-F[p6#F'-F\\o6$7$F`qF(Q$<5>Fco-F[p6#F8-F\\o6$7$FgoF9Q$< 6>Fco-F[p6#FI-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 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" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Another example, for " }{TEXT 283 1 "n" }{TEXT -1 36 " = \+ 210 (a 4-dimensional hypercube:-)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "subZ(210);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6]o-%%TEXTG6$7$$\"\"%\"\"!$!#H!\"\"Q%<21>6\"-%'POINTSG6#7 $$\"\"&F)$!\"$F)-F$6$7$F3F*Q%<35>F.-F06#7$$\"\"\"F)$!\"%F)-F$6$7$F>$!# RF,Q%<30>F.-F06#7$$\"\"#F)F@-F$6$7$FKFEQ%<42>F.-F06#7$$\"\"$F)F@-F$6$7 $FTFEQ%<70>F.-F06#7$F'F@-F$6$7$F'FEQ&<105>F.-F06#7$$\"+++++D!\"*$!\"&F )-F06#7$F>$!\"#F)-F$6$7$F>$!#>F,Q$<2>F.-F06#7$FKFfo-F$6$7$FKF[pQ$<3>F. -F06#7$FTFfo-F$6$7$FTF[pQ$<5>F.-F06#7$F'Ffo-F$6$7$F'F[pQ$<7>F.-F06#7$$ F)F)F5-F$6$7$FfqF*Q$<6>F.-F06#7$F>F5-F$6$7$F>F*Q%<10>F.-F06#7$FKF5-F$6 $7$FKF*Q%<14>F.-F06#7$FTF5-F$6$7$FTF*Q%<15>F.-F06#7$F'F5-F06#7$F^o$F,F )-F$6$7$F^o$F`oF,Q$<1>F.-%'CURVESG6$7$FeqFJ-%'COLOURG6&%$RGBGF)$\"#5F, F)-F]t6$7$FeqF=F`t-F]t6$7$F^qF2F`t-F]t6$7$FgpF2F`t-F]t6$7$FgpF[sF`t-F] t6$7$F^qFbsF`t-F]t6$7$F^qFdrF`t-F]t6$7$F`pFbsF`t-F]t6$7$F`pF[sF`t-F]t6 $7$F`pFeqF`t-F]t6$7$FeoFdrF`t-F]t6$7$FgpF]rF`t-F]t6$7$FesFgpF`t-F]t6$7 $FesF^qF`t-F]t6$7$FesFeoF`t-F]t6$7$FeoF]rF`t-F]t6$7$FeoFeqF`t-F]t6$7$F esF`pF`t-F]t6$7$F=F]oF`t-F]t6$7$FfnF]oF`t-F]t6$7$FSF]oF`t-F]t6$7$FJF]o F`t-F]t6$7$F[sFfnF`t-F]t6$7$FbsFJF`t-F]t6$7$F2FfnF`t-F]t6$7$F2FSF`t-F] t6$7$FbsFfnF`t-F$6$7$F^o$!$3&FgoQ$<0>F.-F]t6$7$F[sF=F`t-F]t6$7$FdrFSF` t-F]t6$7$F]rFSF`t-F]t6$7$F]rF=F`t-F]t6$7$FdrFJF`t-%*AXESSTYLEG6#%%NONE G" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" " Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21 " "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "C urve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40 " "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "C urve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59 " "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Don't stop at that, draw a 5-dimensional \+ hypercube!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subZ(2310);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6]t-%'CURVESG6$7 $7$$\"\"(\"\"!$!\"%F*7$$\"+++++l!\"*$!\"&F*-%'COLOURG6&%$RGBGF*$\"#5! 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Find which of t he groups " }{TEXT -1 1 "U" }{TEXT 288 1 "(" }{TEXT -1 1 "n" }{TEXT 289 7 ") with " }{TEXT -1 1 "n" }{TEXT 290 61 " from 46 to 54 are cycl ic, and find the generators for them. " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 291 1 "2" }{TEXT -1 49 ". Find the number of elements of the order 10 in " }{XPPEDIT 18 0 "Z[20];" "6#&%\"ZG6#\"#?" }{TEXT -1 23 " and list all of them. " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 292 3 "3. " }{TEXT -1 46 "Find the number of elements of the order 2 in " }{TEXT 293 1 "U" }{TEXT -1 6 "(24). " }}{PARA 14 "" 0 "" {TEXT -1 0 " " }{TEXT 300 1 "4" }{TEXT -1 49 ". Find the number of elements of the \+ order 12 in " }{TEXT 301 2 "SL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[6 ];" "6#&%\"ZG6#\"\"'" }{TEXT -1 3 "). " }}{PARA 14 "" 0 "" {TEXT 302 1 "5" }{TEXT -1 48 ". Find the number of elements of the order 2 in " }{TEXT 298 2 "GL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[n];" "6#&%\"ZG6 #%\"nG" }{TEXT -1 6 ") for " }{TEXT 299 1 "n" }{TEXT -1 15 " from 7 to 20. " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 303 1 "6" }{TEXT -1 38 ". List the elements of the order 3 in " }{TEXT 304 2 "SL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[3];" "6#&%\"ZG6#\"\"$" }{TEXT -1 3 "). " }} {PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 305 1 "7" }{TEXT -1 28 ". Draw s ubgroup lattices of " }{XPPEDIT 18 0 "Z[8];" "6#&%\"ZG6#\"\")" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Z[12];" "6#&%\"ZG6#\"#7" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "Z[60];" "6#&%\"ZG6#\"#g" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Z[100];" "6#&%\"ZG6#\"$+\"" }{TEXT -1 2 ". " }}}}}}{MARK "0 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }