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Multiplicative Cyclic Groups by Maple

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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

Procedures

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).

>	C:=proc(n,a);

>	GroupElements,[seq(a^i, i = 0 .. n-1)];

end proc;

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>	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;

proc (n, a) local M, L; L := [seq(a^i, i = 0 .. n-1)]; M := Matrix(n, proc (i, j) options operator, arrow; L[i]*L[j] end proc); GroupTable, algsubs(a^n = 1, M) end proc

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$GroupTable, Matrix(8, 8, {(1, 1) = 1, (1, 2) = a, (1, 3) = a^2, (1, 4) = a^3, (1, 5) = a^4, (1, 6) = a^5, (1, 7) = a^6, (1, 8) = a^7, (2, 1) = a, (2, 2) = a^2, (2, 3) = a^3, (2, 4) = a^4, (2, 5) = a^5, (2, 6) = a^6, (2, 7) = a^7, (2, 8) = 1, (3, 1) = a^2, (3, 2) = a^3, (3, 3) = a^4, (3, 4) = a^5, (3, 5) = a^6, (3, 6) = a^7, (3, 7) = 1, (3, 8) = a, (4, 1) = a^3, (4, 2) = a^4, (4, 3) = a^5, (4, 4) = a^6, (4, 5) = a^7, (4, 6) = 1, (4, 7) = a, (4, 8) = a^2, (5, 1) = a^4, (5, 2) = a^5, (5, 3) = a^6, (5, 4) = a^7, (5, 5) = 1, (5, 6) = a, (5, 7) = a^2, (5, 8) = a^3, (6, 1) = a^5, (6, 2) = a^6, (6, 3) = a^7, (6, 4) = 1, (6, 5) = a, (6, 6) = a^2, (6, 7) = a^3, (6, 8) = a^4, (7, 1) = a^6, (7, 2) = a^7, (7, 3) = 1, (7, 4) = a, (7, 5) = a^2, (7, 6) = a^3, (7, 7) = a^4, (7, 8) = a^5, (8, 1) = a^7, (8, 2) = 1, (8, 3) = a, (8, 4) = a^2, (8, 5) = a^3, (8, 6) = a^4, (8, 7) = a^5, (8, 8) = a^6})$

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2) Element Order Table

eo(a, i ,n) dentes the order of

OT(n,a) is the order table.

>	eo:=proc(a,i,n)

>	n/gcd(n,i);

end proc;

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>	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;

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, LinearAlgebra:-Transpose(Matrix(2, n, [H, L])) end proc

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$OrderTable, Matrix(8, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = a, (2, 2) = 8, (3, 1) = a^2, (3, 2) = 4, (4, 1) = a^3, (4, 2) = 8, (5, 1) = a^4, (5, 2) = 2, (6, 1) = a^5, (6, 2) = 8, (7, 1) = a^6, (7, 2) = 4, (8, 1) = a^7, (8, 2) = 8})$

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3)Element Inverse Table

ei(a, i, n) denotes the inverse of

IT(n,a) is the inverse table.

>	ei:= proc(a,i,n)

a^(n-i);

end proc;

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>	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;

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, LinearAlgebra:-Transpose(Matrix(2, n, [L, H])) end proc

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$InverseTable, Matrix(8, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = a, (2, 2) = a^7, (3, 1) = a^2, (3, 2) = a^6, (4, 1) = a^3, (4, 2) = a^5, (5, 1) = a^4, (5, 2) = a^4, (6, 1) = a^5, (6, 2) = a^3, (7, 1) = a^6, (7, 2) = a^2, (8, 1) = a^7, (8, 2) = a})$

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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;

proc (n, a) numtheory:-phi(n), generators, map(proc (x) options operator, arrow; a^x end proc, select(proc (r) options operator, arrow; is(igcd(r, n) = 1) end proc, [`$`(1 .. n-1)])) end proc

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5) Subgroups

. That is

SG(a, k, n) denotes the subgroup generated by

AS(n,a) is the set all subgroups of C(n,a).

>	SG:=proc(a,k,n)

>	sort(algsubs(a^(n)=1,{seq((a^(k))^(i-1),i=1..(n)/(gcd(n,k)))}));

end proc;

$proc (a, k, n) sort(algsubs(a^n = 1, {seq((a^k)^(i-1), i = 1 .. n/gcd(n, k))})) end proc$

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${1, a^2, a^4, a^6, a^8, a^10, a^12, a^14, a^16}$

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${1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11}$

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>	AS:=proc(n,a)

>	local S; S:= {seq(SG(a,t,n),t=0..n-1)};

>	S, print( nops(S) ,subgroups);

end proc;

proc (n, a) local S; S := {seq(SG(a, t, n), t = 0 .. n-1)}; S, print(nops(S), subgroups) end proc

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${{1}, {1, a^4}, {1, a^2, a^4, a^6}, {1, a, a^2, a^3, a^4, a^5, a^6, a^7}}$

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${{1}, {1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12}}$

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${{1}, {1, a^8}, {1, a^4, a^8, a^12}, {1, a^2, a^4, a^6, a^8, a^10, a^12, a^14}, {1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15}}$

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<
"a^(r)> intersect <a^(s)>=<a^([r,s] )> and <<a^(r)> union <a^(s)>>=<a^((r,s))>, where [ ] and ( ) are the notations of lcm and gcd respectively ."

There is a 1-1 correspondence between the lattice L of all subgroups of with order 4 and the set The correspondence is given by S↔ |S| for any subgroup S of

"Thus Hasse diagrams of L and D[n] are similar. The label on each vertix is the order of a subgroup. The diagram can be adjusted by "

suitable rotations.

>	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;

>	G:=Digraph(VV,L);

>	DrawGraph(G,style=spring);

end proc;

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7) Group Homomorphisms

" A group homomorphism f :C[m]<a>-> C[n]<b> is given by f(a)=b^(r), where (n)/((r,n))| (m,n) to gurentee that f is well defined."

" If m=n and (r,n)=1, then f is an automorphism. The automorphism group Aut(C[n])&cong;C[n]^(*),the group of all reduced residue classes, modulo n. It is abelian but not necessarily cyclic. "

Hom(m,a,n,b) denotes the set of all all group homomorphisms

Aut(n,a) is the set of all automorphisms of

>	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;

proc (m, a, n, b) local L; L := select(proc (x) options operator, arrow; is(igcd(m, n)*igcd(x, n)/n, integer) end proc, [`$`(0 .. n-1)]); 'f(a)' = seq(b^(`mod`(r, n)), `in`(r, L)), nops(L), homomorphisms end proc

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>	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;

proc (n, a) local L; L := select(proc (x) options operator, arrow; is(igcd(x, n) = 1) end proc, [`$`(1 .. n)]); 'f(a)' = seq(a^r, `in`(r, L)), nops(L), automorphisms end proc

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8) Kernel and Image

A group homomorphisms

The following procedures are to check that

First isomorphism theorem implies that |Ker| |Ima|=|G|.

>	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;

proc (m, a, n, b, r) if is(igcd(m, n)*igcd(r, n)/n, integer) then print(true) else print(false) end if end proc

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>	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;

$proc (m, a, n, b, r) local K; K := select(proc (x) options operator, arrow; is(`mod`(r*x, n) = 0) end proc, {`$`(0 .. m-1)}); {seq(a^(`mod`(y, m)), `in`(y, K))} end proc$