Multiplicative Cyclic Groups by Maple Kahtan H. Alzubaidy Department of Mathematics, Faculty of Science, University of Benghazi E-mail: kahtanalzubaidy@yahoo.com Introduction In this article we introduce cyclic groups in multiplicative notations by using Maple 13. Group table,order table,and inverse table are given. All generators and all subgroups of a multiplicative cyclic group are given also. Hasse diagrams of subgroup lattices are shown. All homomorphisms and automorphisms of such groups are computed as well as the kernel and image of a homomorphism. Reference 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 Procedures LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 1) Group Table C(n,a) denotes a multiplicative cyclic group of order n and generator a . GT(n,a) is the group table of C(n,a). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2J1EocmVzdGFydEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUSI7RidGLy9GNUY5RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGLy8lK2V4ZWN1dGFibGVHRkFGPkY3 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 JSFH C:=proc(n,a); GroupElements,[seq(a^i, i = 0 .. n-1)]; end proc; Zio2JEkibkc2IkkiYUdGJUYlRiVGJTYkSS5Hcm91cEVsZW1lbnRzR0YlNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQpRiZJImlHRiUvRi87IiIhLCZGJCIiIkY0ISIiRiVGJUYl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 NiRJLkdyb3VwRWxlbWVudHNHNiI3KiIiIkkiYUdGJCokKUYnIiIjRiYqJClGJyIiJEYmKiQpRiciIiVGJiokKUYnIiImRiYqJClGJyIiJ0YmKiQpRiciIihGJg== JSFH GT:=proc(n,a) local M,L ;L:=[seq(a^i, i =0..n-1)]; M:=Matrix(n,(i,j)-> L[i]*L[j]); GroupTable,algsubs(a^n=1,M); end proc; Zio2JEkibkc2IkkiYUdGJTYkSSJNR0YlSSJMR0YlRiVGJUMlPkYpNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQpRiZJImlHRiUvRjI7IiIhLCZGJCIiIkY3ISIiPkYoLUknTWF0cml4R0YlNiRGJGYqNiRGMkkiakdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYmRik2I0YyRjcmRik2I0Y/RjdGJUYlNiRGKTglNiRJK0dyb3VwVGFibGVHRiUtSShhbGdzdWJzR0YlNiQvKUYmRiRGN0YoRiVGJUYl JSFH 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 2) Element Order Table eo(a, i ,n) dentes the order of 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 OT(n,a) is the order table. eo:=proc(a,i,n) n/gcd(n,i); end proc; Zio2JUkiYUc2IkkiaUdGJUkibkdGJUYlRiVGJSomRiciIiItSSRnY2RHRiU2JEYnRiYhIiJGJUYlRiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 IiIm 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 IiIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= OT:=proc(n,a) local L,H;L:=[seq(eo(a,i,n),i=0..n-1)];H:=[seq(a^i, i =0..n-1)]; OrderTable,Transpose(Matrix(2,n,[H,L])); end proc; Zio2JEkibkc2IkkiYUdGJTYkSSJMR0YlSSJIR0YlRiVGJUMlPkYoNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSNlb0dGJTYlRiZJImlHRiVGJC9GNDsiIiEsJkYkIiIiRjkhIiI+Rik3Iy1GLjYkKUYmRjRGNTYkSStPcmRlclRhYmxlR0YlLV9JLkxpbmVhckFsZ2VicmFHNiRGL0koX3N5c2xpYkdGJUkqVHJhbnNwb3NlR0YlNiMtSSdNYXRyaXhHRkU2JSIiI0YkNyRGKUYoRiVGJUYl JSFH 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 3)Element Inverse Table ei(a, i, n) denotes the inverse of 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 IT(n,a) is the inverse table. ei:= proc(a,i,n) a^(n-i); end proc; Zio2JUkiYUc2IkkiaUdGJUkibkdGJUYlRiVGJSlGJCwmRiciIiJGJiEiIkYlRiVGJQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 KiQpSSJiRzYiIiImIiIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= IT:=proc(n,a) local L,H; L:=[seq(a^i, i =0..n-1)]; H:=algsubs(a^(n)=1,[seq(a^(n-i),i=0..n-1)]); InverseTable,Transpose(Matrix(2,n,[L,H])); end proc; Zio2JEkibkc2IkkiYUdGJTYkSSJMR0YlSSJIR0YlRiVGJUMlPkYoNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQpRiZJImlHRiUvRjI7IiIhLCZGJCIiIkY3ISIiPkYpLUkoYWxnc3Vic0c2JEYvSShfc3lzbGliR0YlNiQvKUYmRiRGNzcjLUYuNiQpRiYsJkYkRjdGMkY4RjM2JEktSW52ZXJzZVRhYmxlR0YlLV9JLkxpbmVhckFsZ2VicmFHRjxJKlRyYW5zcG9zZUdGJTYjLUknTWF0cml4R0Y8NiUiIiNGJDckRihGKUYlRiVGJQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 4) Generators AG(n,a) denotes all generators of C(n,a). AG:=proc(n,a) phi(n), generators, map(x->a^x,select(r->is(igcd(r,n)=1),[$1..n-1])); end proc; Zio2JEkibkc2IkkiYUdGJUYlRiVGJTYlLV9JKm51bXRoZW9yeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUkkcGhpR0YlNiNGJEkrZ2VuZXJhdG9yc0dGJS1JJG1hcEdGLDYkZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKUYmRjZGJUYlNiRGJjklLUknc2VsZWN0R0YsNiRmKjYjSSJyR0YlRiVGN0YlLUkjaXNHRis2Iy8tSSVpZ2NkR0YsNiRGQkYkIiIiRiVGJTYkRiQ5JDcjLUkiJEdGLDYjO0ZKLCZGJEZKRkohIiJGJUYlRiU= JSFH 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 NiUiIiVJK2dlbmVyYXRvcnNHNiI3JkkiYUdGJSokKUYnIiIkIiIiKiQpRiciIihGKyokKUYnIiIqRis= 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 NiUiIidJK2dlbmVyYXRvcnNHNiI3KEkieEdGJSokKUYnIiImIiIiKiQpRiciIihGKyokKUYnIiM2RisqJClGJyIjOEYrKiQpRiciIzxGKw== 5) Subgroups 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY9LUkjbW9HRiQ2MFEifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRktGK0YrRitGK0YrLUkjbWlHRiQ2KFEmRXZlcnlGJ0YvRjIvJSdpdGFsaWNHRjxGNUY4RistRk82KFEpcXVvdGllbnRGJ0YvRjJGUkY1RjhGKy1GTzYoUSZncm91cEYnRi9GMkZSRjVGOEYrLUZPNihRI29mRidGL0YyRlJGNUY4RistRk82KFEiYUYnRi9GMkZSRjVGOEYrLUZPNihRJ2N5Y2xpY0YnRi9GMkZSRjVGOEYrRldGKy1GTzYoUSNpc0YnRi9GMkZSRjVGOEYrRmpuRi9GMkY1Rjg=. That is 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 SG(a, k, n) denotes the subgroup generated by 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 AS(n,a) is the set all subgroups of C(n,a). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= SG:=proc(a,k,n) sort(algsubs(a^(n)=1,{seq((a^(k))^(i-1),i=1..(n)/(gcd(n,k)))})); end proc; Zio2JUkiYUc2Ikkia0dGJUkibkdGJUYlRiVGJS1JJXNvcnRHJSpwcm90ZWN0ZWRHNiMtSShhbGdzdWJzRzYkRipJKF9zeXNsaWJHRiU2JC8pRiRGJyIiIjwjLUkkc2VxR0YqNiQpKUYkRiYsJkkiaUdGJUYzRjMhIiIvRjs7RjMqJkYnRjMtSSRnY2RHRi42JEYnRiZGPEYlRiVGJQ== JSFH 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 PCsiIiIqJClJImFHNiIiIiNGIyokKUYmIiIlRiMqJClGJiIiJ0YjKiQpRiYiIilGIyokKUYmIiM1RiMqJClGJiIjN0YjKiQpRiYiIzlGIyokKUYmIiM7RiM= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2J1EjU0dGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjEvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkobWZlbmNlZEdGJDYmLUYjNiotRiw2J1EiYUYnRi9GMkY0RjctSSNtb0dGJDYvUSIsRidGLy9GNUY5RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtbkdGJDYmUSIwRidGL0ZGRjdGQi1GZ242JlEjMTJGJ0YvRkZGN0YvRkZGN0YvRkZGNy1GQzYvUSI7RidGL0ZGRjdGR0ZKRkxGTkZQRlJGVEZWL0ZaUSwwLjI3Nzc3NzhlbUYnRi8vJStleGVjdXRhYmxlR0ZJRkZGNw== PCMiIiI= 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 PC4iIiJJImFHNiIqJClGJCIiI0YjKiQpRiQiIiRGIyokKUYkIiIlRiMqJClGJCIiJkYjKiQpRiQiIidGIyokKUYkIiIoRiMqJClGJCIiKUYjKiQpRiQiIipGIyokKUYkIiM1RiMqJClGJCIjNkYj LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= AS:=proc(n,a) local S; S:= {seq(SG(a,t,n),t=0..n-1)}; S, print( nops(S) ,subgroups); end proc; Zio2JEkibkc2IkkiYUdGJTYjSSJTR0YlRiVGJUMkPkYoPCMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSNTR0dGJTYlRiZJInRHRiVGJC9GMzsiIiEsJkYkIiIiRjghIiI2JEYoLUkmcHJpbnRHRi42JC1JJW5vcHNHRi5GJ0kqc3ViZ3JvdXBzR0YlRiVGJUYl 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 NiQiIiVJKnN1Ymdyb3Vwc0c2Ig== PCY8IyIiIjwkRiQqJClJImFHNiIiIiVGJDwmRiQqJClGKCIiI0YkRiYqJClGKCIiJ0YkPCpGJEYoRiwqJClGKCIiJEYkRiYqJClGKCIiJkYkRi8qJClGKCIiKEYk LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 NiQiIiJJKnN1Ymdyb3Vwc0c2Ig== PCM8IyIiIg== 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 NiQiIiNJKnN1Ymdyb3Vwc0c2Ig== PCQ8IyIiIjwvRiRJImFHNiIqJClGJiIiI0YkKiQpRiYiIiRGJCokKUYmIiIlRiQqJClGJiIiJkYkKiQpRiYiIidGJCokKUYmIiIoRiQqJClGJiIiKUYkKiQpRiYiIipGJCokKUYmIiM1RiQqJClGJiIjNkYkKiQpRiYiIzdGJA== 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 NiQiIiZJKnN1Ymdyb3Vwc0c2Ig== PCc8IyIiIjwkRiQqJClJImFHNiIiIilGJDwmRiQqJClGKCIiJUYkRiYqJClGKCIjN0YkPCpGJCokKUYoIiIjRiRGLCokKUYoIiInRiRGJiokKUYoIiM1RiRGLyokKUYoIiM5RiQ8MkYkRihGMyokKUYoIiIkRiRGLCokKUYoIiImRiRGNiokKUYoIiIoRiRGJiokKUYoIiIqRiRGOSokKUYoIiM2RiRGLyokKUYoIiM4RiRGPCokKUYoIiM6RiQ= 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 <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 There is a 1-1 correspondence between the lattice L of all subgroups of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2KFEiQ0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y6LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GIzYkLUYsNihRIm5GJ0Y1RjhGO0Y9RkAvRj5RJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGSA== with order \342\212\206 and the set LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2KFEiREYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGQC1GIzYkLUYsNihRIm5GJ0Y1RjgvRjxGOi9GP1EsYm9sZC1pdGFsaWNGJ0ZBL0Y/USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRks=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The correspondence is given by S\342\206\224 |S| for any subgroup S of 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 suitable rotations. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Hasse:=proc(n) local V,VV,sdiv,r,nc,L,x,y,G; V:=divisors(n); VV:=convert(V,list); sdiv:=(x,y)->(x<>y) and is (y/x,integer); r:=(x,z,y)->is(sdiv(x,z) and sdiv(z,y)); nc:=(x,y)->convert({ seq(not(r(x,z,y)),z in V)},list)[1]; L:={}; for x in V do for y in V do if (sdiv(x,y) and nc(x,y)) then L:=L union {[x,y]}fi od od; L; G:=Digraph(VV,L); DrawGraph(G,style=spring); end proc; 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 7) Group Homomorphisms 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 Hom(m,a,n,b) denotes the set of all all group homomorphisms 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 Aut(n,a) is the set of all automorphisms of 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 Hom:=proc(m,a,n,b) local L; L:=select(x->is(igcd(m,n)*igcd(x,n)/n ,integer),[$0..n-1]); 'f(a)'=seq(b^(r mod n),r in L), nops(L),homomorphisms; end proc; Zio2JkkibUc2IkkiYUdGJUkibkdGJUkiYkdGJTYjSSJMR0YlRiVGJUMkPkYqLUknc2VsZWN0RyUqcHJvdGVjdGVkRzYkZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjaXNHNiRGL0koX3N5c2xpYkdGJTYkKigtSSVpZ2NkR0YvNiRGJEYnIiIiLUY+NiRGM0YnRkBGJyEiIkkoaW50ZWdlckdGL0YlRiU2JkYkOSRGJzkmNyMtSSIkR0YvNiM7IiIhLCZGJ0ZARkBGQzYlLy4tSSJmR0YlNiNGJi1JJHNlcUdGLzYkKUYoLUkkbW9kR0YlNiRJInJHRiVGJy1JI2luR0YvNiRGZm5GKi1JJW5vcHNHRi9GKUkuaG9tb21vcnBoaXNtc0dGJUYlRiVGJQ== JSFH 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 NiUvLUkiZkc2IjYjSSJhR0YmNiMiIiJGKkkuaG9tb21vcnBoaXNtc0dGJg== 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 NiUvLUkiZkc2IjYjSSJhR0YmNiMiIiJGKkkuaG9tb21vcnBoaXNtc0dGJg== 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 NiUvLUkiZkc2IjYjSSJhR0YmNi4iIiIqJClJImJHRiYiIiRGKiokKUYtIiInRioqJClGLSIiKkYqKiQpRi0iIzdGKiokKUYtIiM6RioqJClGLSIjPUYqKiQpRi0iI0BGKiokKUYtIiNDRioqJClGLSIjRkYqKiQpRi0iI0lGKiokKUYtIiNMRipGN0kuaG9tb21vcnBoaXNtc0dGJg== 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 NiUvLUkiZkc2IjYjSSJhR0YmNi4iIiIqJClJImJHRiYiIiNGKiokKUYtIiIlRioqJClGLSIiJ0YqKiQpRi0iIilGKiokKUYtIiM1RioqJClGLSIjN0YqKiQpRi0iIzlGKiokKUYtIiM7RioqJClGLSIjPUYqKiQpRi0iIz9GKiokKUYtIiNBRipGPUkuaG9tb21vcnBoaXNtc0dGJg== 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 NiUvLUkiZkc2IjYjSSJhR0YmNiciIiJJImJHRiYqJClGKyIiI0YqKiQpRisiIiRGKiokKUYrIiIlRioiIiZJLmhvbW9tb3JwaGlzbXNHRiY= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Aut:=proc(n,a) local L;L:=select(x->is (igcd(x,n)=1),[$1..n]); 'f(a)'=seq(a^r,r in L), nops(L),automorphisms; end proc; Zio2JEkibkc2IkkiYUdGJTYjSSJMR0YlRiVGJUMkPkYoLUknc2VsZWN0RyUqcHJvdGVjdGVkRzYkZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjaXNHNiRGLUkoX3N5c2xpYkdGJTYjLy1JJWlnY2RHRi02JEYxRiQiIiJGJUYlNiRGJDkkNyMtSSIkR0YtNiM7Rj5GJDYlLy4tSSJmR0YlNiNGJi1JJHNlcUdGLTYkKUYmSSJyR0YlLUkjaW5HRi02JEZQRigtSSVub3BzR0YtRidJLmF1dG9tb3JwaGlzbXNHRiVGJUYlRiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 NiUvLUkiZkc2IjYjSSJhR0YmNihGKCokKUYoIiIjIiIiKiQpRigiIiRGLSokKUYoIiIlRi0qJClGKCIiJkYtKiQpRigiIidGLUY5SS5hdXRvbW9ycGhpc21zR0Ym 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 NiUvLUkiZkc2IjYjSSJ4R0YmNiZGKCokKUYoIiIkIiIiKiQpRigiIihGLSokKUYoIiIqRi0iIiVJLmF1dG9tb3JwaGlzbXNHRiY= 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 NiUvLUkiZkc2IjYjSSJ4R0YmNipGKCokKUYoIiIoIiIiKiQpRigiIzZGLSokKUYoIiM4Ri0qJClGKCIjPEYtKiQpRigiIz5GLSokKUYoIiNCRi0qJClGKCIjSEYtIiIpSS5hdXRvbW9ycGhpc21zR0Ym 8) Kernel and Image A group homomorphisms 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZIRistSSNtaUdGJDYjUSFGJ0YvLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0YyRjU= 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 The following procedures are to check that 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 First isomorphism theorem implies that |Ker| |Ima|=|G|. JSFH CHK:=proc(m,a,n,b,r) if is(igcd(m,n)*igcd(r,n)/n,integer) then print(true)else print(false) fi;end proc; Zio2J0kibUc2IkkiYUdGJUkibkdGJUkiYkdGJUkickdGJUYlRiVGJUAlLUkjaXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JCooLUklaWdjZEdGLjYkRiRGJyIiIi1GMzYkRilGJ0Y1RichIiJJKGludGVnZXJHRi4tSSZwcmludEdGLjYjSSV0cnVlR0YuLUY7NiNJJmZhbHNlR0YuRiVGJUYl JSFH 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 SSV0cnVlRyUqcHJvdGVjdGVkRw== 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 SSZmYWxzZUclKnByb3RlY3RlZEc= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Ker:=proc(m,a,n,b,r) local K; K:=select(x->is(r*x mod n =0),{$0..m-1}); {seq(a^(y mod m),y in K)}; end proc; Zio2J0kibUc2IkkiYUdGJUkibkdGJUkiYkdGJUkickdGJTYjSSJLR0YlRiVGJUMkPkYrLUknc2VsZWN0RyUqcHJvdGVjdGVkRzYkZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjaXNHNiRGMEkoX3N5c2xpYkdGJTYjLy1JJG1vZEdGJTYkKiZGKSIiIkY0RkJGJyIiIUYlRiU2JkYpOShGJzkmPCMtSSIkR0YwNiM7RkMsJkYkRkJGQiEiIjwjLUkkc2VxR0YwNiQpRiYtRj82JEkieUdGJUYkLUkjaW5HRjA2JEZVRitGJUYlRiU= JSFH Ima:=proc(m,a,n,b,r) local M; M:={seq( r*t mod n, t=0..m-1)}; {seq(b^x , x in M)}; end proc; Zio2J0kibUc2IkkiYUdGJUkibkdGJUkiYkdGJUkickdGJTYjSSJNR0YlRiVGJUMkPkYrPCMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSRtb2RHRiU2JComRikiIiJJInRHRiVGN0YnL0Y4OyIiISwmRiRGN0Y3ISIiPCMtRjA2JClGKEkieEdGJS1JI2luR0YxNiRGQkYrRiVGJUYl JSFH Examples 1) H(18,a,12,b,4) 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 SSV0cnVlRyUqcHJvdGVjdGVkRw== 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 NiQ8KCIiIiokKUkiYUc2IiIiJEYkKiQpRiciIidGJCokKUYnIiIqRiQqJClGJyIjN0YkKiQpRiciIzpGJDwlRiQqJClJImJHRigiIiVGJCokKUY5IiIpRiQ= JSFH 2) H(12,a,18,b,4) 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 SSV0cnVlRyUqcHJvdGVjdGVkRw== 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 NiQ8JCIiIiokKUkiYUc2IiIiJ0YkPChGJCokKUkiYkdGKCIiJEYkKiQpRi1GKUYkKiQpRi0iIipGJCokKUYtIiM3RiQqJClGLSIjOkYk JSFH 3) H(10,a,10,a,7); 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 SSV0cnVlRyUqcHJvdGVjdGVkRw== 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 NiQ8IyIiIjwsRiRJImFHNiIqJClGJiIiI0YkKiQpRiYiIiRGJCokKUYmIiIlRiQqJClGJiIiJkYkKiQpRiYiIidGJCokKUYmIiIoRiQqJClGJiIiKUYkKiQpRiYiIipGJA== JSFH TTdSMApJNlJUQUJMRV9TQVZFLzE1MzE0MTI4NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhI1tvIikiKSIiIiUiYUcqJEYoIiIjKiRGKCIiJCokRigiIiUqJEYoIiImKiRGKCIiJyokRigiIihGKEYpRitGLUYvRjFGM0YnRilGK0YtRi9GMUYzRidGKEYrRi1GL0YxRjNGJ0YoRilGLUYvRjFGM0YnRihGKUYrRi9GMUYzRidGKEYpRitGLUYxRjNGJ0YoRilGK0YtRi9GM0YnRihGKUYrRi1GL0YxNiI=TTdSMApJNlJUQUJMRV9TQVZFLzE3MTk4NTQ2OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiKSIjIiIiJSJhRyokRigiIiMqJEYoIiIkKiRGKCIiJSokRigiIiYqJEYoIiInKiRGKCIiKEYnIiIpRi5GNUYqRjVGLkY1NiI=TTdSMApJNlJUQUJMRV9TQVZFLzE3MTk4NjE3MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIzEiKSIjIiIiJSJhRyokRigiIiMqJEYoIiIkKiRGKCIiJSokRigiIiYqJEYoIiInKiRGKCIiKEYnRjNGMUYvRi1GK0YpRig2Ig==