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Metacyclic Groups by Maple

Kahtan H. Alzubaidy

Department of Mathematics, Faculty of Science,University of Benghazi,Liya

E-mail: kahtanalzubaidy@yahoo.com

 

 

 

 

Key Words : Metacyclic Groups, Recurrence Relations

 

Introduction.

 A group G is called metacyclic if it has a normal subgroup H such that both H and G/H are cyclic. A metacyclic group is presented by two generators and three relations.

An important class of metacyclic groups is finitely presented as follows

G=<a,b: "a^(m)=b^(n)=1,ba=a^(r)b,r^(n)&equiv;1 (mod m)>. Thus "G={ "a^(i)b^(j) : 0<=i<m, 0<=j<n }. |G|=mn."

G y "C[m]<a>&rtimes; C[n]<b> as a semidirect product."

If  r =1, then G y "C[m]<a>* C[n]<b> which is abelian."

In this article a few procedures are made to compute the list of group elements, group table, a table of the inverse elements, and a table of the order of the elements for the metacylic group G.The centralizer and center are computed. Finally conjugate classes are also found.

A finite metacyclic group of this class is represented by a finite group of complex numbers with a special binary operation.Complex numbers in this respect simplifies Maple computation instead of dealing with powerw of the generators a and b directly.

The advantige of the approach of this application is that mathematical forms are maintained so that the results look like exactly as in mathematics.

 

Referencs

[1] Michael O. Albertson,Joan P. Hutchinson, Discrete Mathematics with Algorithms, John Wiley & Sons,New York, 1988.  

[2] Thomas W. Hungerford, Algebra, Springer-Verlag, Graduate Texts in Mathematics 73, New York, 1974.

 

Computations

 

> 

restart

> 

with(numtheory):

> 

NULL

The following procedure determine the values of r if m and n are given.

 

> 

r:= proc(m,n) local L,x;L:={};

> 

 for x from 2 to m-1 do if is(x^n mod m =1)then L:=L union {x} fi od;min(L);

> 

end proc;

proc (m, n) local L, x; L := {}; for x from 2 to m-1 do if is(`mod`(x^n, m) = 1) then L := `union`(L, {x}) end if end do; min(L) end proc

(1)
> 

NULL

> 

r(4, 2)

3

(2)
> 

r(7, 3)

2

(3)
> 

``

 We have taken the minimum value of  r not 1. The other values may produce isomorphic copies of groups.To get all values of r

> 

msolve(x^3=1,7);

 

{x = 1}, {x = 2}, {x = 4}

(4)

 

Notation: Our group is denoted by  G(a,m,b,n,r).

The following procedure lists the elements of the group  G(a,m,b,n,r).

 

> 

G:=proc(a,m,b,n,r)

> 

 local M;

> 

 M:=Transpose(Matrix(m,n,(i,j)->a^(i-1)*b^(j-1)));

> 

 [seq(seq(M(i,j),j=1..m),i=1..n)], m*n, elements;

> 

end proc;

proc (a, m, b, n, r) local M; M := LinearAlgebra:-Transpose(Matrix(m, n, proc (i, j) options operator, arrow; a^(i-1)*b^(j-1) end proc)); [seq(seq(M(i, j), j = 1 .. m), i = 1 .. n)], m*n, elements end proc

(5)
> 

NULL

> 

G(a, 4, b, 2, 3)

[1, a, a^2, a^3, b, a*b, a^2*b, a^3*b], 8, elements

(6)
> 

G(x, 5, y, 4, 2)

[1, x, x^2, x^3, x^4, y, x*y, x^2*y, x^3*y, x^4*y, y^2, x*y^2, x^2*y^2, x^3*y^2, x^4*y^2, y^3, x*y^3, x^2*y^3, x^3*y^3, x^4*y^3], 20, elements

(7)

 

 

 From ba = a^r*b we have  b^j*a^k = Typesetting[delayDotProduct](a^(k*r^j)*b^j.Thus, group, true)*operation*can*be*viewed*as*follows

 

("a^(i)b^(j))(a^(k)b^(l))=a^(i+k r^(j) (mod m) )b^(j+l (mod n))."

 

The identity is  
"a^(0)b^(0)=1 and the inverse element is given by (a^(i) b^(j))^(-1)= a^((m-i) r^((n-j)) (mod m) )b^((n-j) (mod n)) ."

 

Complex representation

 

 We make a complex copy GC of the metacyclic group G.

 

"a^(i)b^(j)->i+j*I, where I=sqrt(-1) and z-> a^(Re(z)) b^(Im(z)) . Thus group elements in complex form are given by GC= {i+j*I: 0<=i<m, 0<=j<n }."

 

Thus the operation, identity, and inverse element in GC are given as follows

p(z,w)=(Re(z)+"r^(Im(z)).Re(w)) (mod m)+(Im(z)+Im(w)) (mod n)*I"

 

identity 0+0*I =0

 

inv(z)= (m - Re(z))*"r^(n-Im(z)) (mod m)+(n-Im(z)) (mod n) *I ."

 

Group elements of the complex copy GC  are given by the following procedure

 

 

> 

GC:=proc(a,m,b,n,r)

> 

 local M;

> 

 M:=Transpose(Matrix(m,n,(i,j)->Complex(i-1,j-1)));

> 

 [seq(seq(M(i,j),j=1..m),i=1..n)];

> 

end proc;

proc (a, m, b, n, r) local M; M := LinearAlgebra:-Transpose(Matrix(m, n, proc (i, j) options operator, arrow; Complex(i-1, j-1) end proc)); [seq(seq(M(i, j), j = 1 .. m), i = 1 .. n)] end proc

(8)
> 

``

> 

GC(a, 3, b, 2, 2); nops(%), element*s

[0, 1, 2, I, 1+I, 2+I]

6, element*s

(9)
> 

GC(x, 5, y, 4, 2); nops(%), elements

[0, 1, 2, 3, 4, I, 1+I, 2+I, 3+I, 4+I, 2*I, 1+2*I, 2+2*I, 3+2*I, 4+2*I, 3*I, 1+3*I, 2+3*I, 3+3*I, 4+3*I]

20, elements

(10)
> 

``

 

Group Tables

 

We start first with the complex group table GTC of the group GC.

 

> 

T:=proc(a,m,b,n,r)NULL

> 

local L,p;

> 

p := proc (z, w) options operator, arrow; `mod`(Re(z)+r^Im(z)*Re(w), m)+I*(`mod`(Im(z)+Im(w), n)) end proc;

> 

L:=GC(a,m,b,n,r);

> 

Matrix(m*n,(i,j)->p(L[i],L[j]));

> 

end proc;

proc (a, m, b, n, r) local L, p; p := proc (z, w) options operator, arrow; `mod`(Re(z)+r^Im(z)*Re(w), m)+I*(`mod`(Im(z)+Im(w), n)) end proc; L := GC(a, m, b, n, r); Matrix(m*n, proc (i, j) options operator, arrow; p(L[i], L[j]) end proc) end proc

(11)
> 

``

> 

``

> 

GTC:=proc(a,m,b,n,r)

> 

local L,R,RR,P,N;

> 

 

> 

L:=GC(a,m,b,n,r);

> 

R:=convert([L],matrix);

> 

RR:=transpose(R);N:=T(a,m,b,n,r);

> 

P:=Matrix([[omicron]]);

> 

blockmatrix(2,2,[P,R,RR,N,P,RR,R,N]);

> 

end proc;

proc (a, m, b, n, r) local L, R, RR, P, N; L := GC(a, m, b, n, r); R := convert([L], linalg:-matrix); RR := linalg:-transpose(R); N := T(a, m, b, n, r); P := Matrix([[omicron]]); linalg:-blockmatrix(2, 2, [P, R, RR, N, P, RR, R, N]) end proc

(12)
> 

``

> 

GTC(a, 3, b, 2, 2)

array( 1 .. 7, 1 .. 7, [( 3, 4 ) = (0), ( 2, 2 ) = (0), ( 6, 6 ) = (0), ( 3, 6 ) = (2+I), ( 1, 4 ) = (2), ( 6, 3 ) = (I), ( 4, 5 ) = (2+I), ( 7, 1 ) = (2+I), ( 4, 6 ) = (I), ( 5, 5 ) = (0), ( 4, 1 ) = (2), ( 7, 5 ) = (2), ( 3, 3 ) = (2), ( 6, 5 ) = (1), ( 4, 3 ) = (0), ( 7, 6 ) = (1), ( 1, 3 ) = (1), ( 2, 5 ) = (I), ( 2, 7 ) = (2+I), ( 5, 4 ) = (1+I), ( 1, 7 ) = (2+I), ( 2, 6 ) = (1+I), ( 7, 2 ) = (2+I), ( 5, 7 ) = (1), ( 4, 7 ) = (1+I), ( 6, 4 ) = (2+I), ( 2, 3 ) = (1), ( 1, 5 ) = (I), ( 5, 6 ) = (2), ( 1, 1 ) = (omicron), ( 6, 1 ) = (1+I), ( 6, 2 ) = (1+I), ( 4, 2 ) = (2), ( 1, 2 ) = (0), ( 7, 7 ) = (0), ( 2, 4 ) = (2), ( 3, 2 ) = (1), ( 7, 4 ) = (I), ( 6, 7 ) = (2), ( 4, 4 ) = (1), ( 5, 3 ) = (2+I), ( 5, 2 ) = (I), ( 3, 1 ) = (1), ( 3, 7 ) = (I), ( 5, 1 ) = (I), ( 2, 1 ) = (0), ( 7, 3 ) = (1+I), ( 3, 5 ) = (1+I), ( 1, 6 ) = (1+I) ] )

(13)
> 

GTC(a, 4, b, 2, 3)

array( 1 .. 9, 1 .. 9, [( 9, 3 ) = (2+I), ( 8, 9 ) = (3), ( 5, 7 ) = (I), ( 8, 3 ) = (1+I), ( 6, 3 ) = (3+I), ( 9, 2 ) = (3+I), ( 4, 3 ) = (3), ( 4, 7 ) = (3+I), ( 1, 4 ) = (2), ( 6, 8 ) = (2), ( 2, 2 ) = (0), ( 3, 7 ) = (2+I), ( 2, 7 ) = (1+I), ( 8, 5 ) = (3+I), ( 3, 9 ) = (I), ( 4, 2 ) = (2), ( 3, 2 ) = (1), ( 6, 7 ) = (3), ( 2, 6 ) = (I), ( 6, 9 ) = (1), ( 3, 8 ) = (3+I), ( 7, 7 ) = (0), ( 3, 3 ) = (2), ( 7, 1 ) = (1+I), ( 6, 4 ) = (2+I), ( 9, 9 ) = (0), ( 4, 8 ) = (I), ( 3, 6 ) = (1+I), ( 1, 3 ) = (1), ( 1, 1 ) = (omicron), ( 4, 6 ) = (2+I), ( 1, 5 ) = (3), ( 9, 5 ) = (I), ( 8, 1 ) = (2+I), ( 5, 6 ) = (3+I), ( 2, 9 ) = (3+I), ( 1, 7 ) = (1+I), ( 4, 1 ) = (2), ( 7, 3 ) = (I), ( 8, 8 ) = (0), ( 4, 5 ) = (1), ( 9, 4 ) = (1+I), ( 7, 2 ) = (1+I), ( 4, 9 ) = (1+I), ( 5, 1 ) = (3), ( 2, 1 ) = (0), ( 8, 6 ) = (2), ( 2, 8 ) = (2+I), ( 5, 3 ) = (0), ( 5, 8 ) = (1+I), ( 7, 5 ) = (2+I), ( 8, 4 ) = (I), ( 2, 5 ) = (3), ( 5, 2 ) = (3), ( 1, 6 ) = (I), ( 7, 9 ) = (2), ( 1, 2 ) = (0), ( 3, 1 ) = (1), ( 9, 7 ) = (2), ( 6, 6 ) = (0), ( 9, 1 ) = (3+I), ( 2, 3 ) = (1), ( 6, 2 ) = (I), ( 9, 8 ) = (1), ( 8, 7 ) = (1), ( 8, 2 ) = (2+I), ( 6, 1 ) = (I), ( 2, 4 ) = (2), ( 3, 5 ) = (0), ( 7, 8 ) = (3), ( 1, 8 ) = (2+I), ( 5, 4 ) = (1), ( 5, 5 ) = (2), ( 9, 6 ) = (3), ( 5, 9 ) = (2+I), ( 6, 5 ) = (1+I), ( 4, 4 ) = (0), ( 7, 4 ) = (3+I), ( 7, 6 ) = (1), ( 1, 9 ) = (3+I), ( 3, 4 ) = (3) ] )

(14)

 

Now, we compute  the group table GT of  G in terms of the usual notations of our metacyclic group.

 

> 

h:=proc(w,a,b)

> 

  if w=omicron then omicron else a^Re(w)*b^Im(w)  fi;

> 

end proc;

proc (w, a, b) if w = omicron then omicron else a^Re(w)*b^Im(w) end if end proc

(15)
> 

``

> 

GT:=proc(a,m,b,n,r)

> 

local M,h1; M:=GTC(a,m,b,n,r); h1:=w->h(w,a,b);

> 

Map(h1,M);

> 

 

> 

end proc;

proc (a, m, b, n, r) local M, h1; M := GTC(a, m, b, n, r); h1 := proc (w) options operator, arrow; h(w, a, b) end proc; LinearAlgebra:-Map(h1, M) end proc

(16)
> 

``

> 

``

> 

GT(a, 3, b, 2, 2)

array( 1 .. 7, 1 .. 7, [( 3, 4 ) = (1), ( 2, 2 ) = (1), ( 6, 6 ) = (1), ( 3, 6 ) = (a^2*b), ( 1, 4 ) = (a^2), ( 6, 3 ) = (b), ( 4, 5 ) = (a^2*b), ( 7, 1 ) = (a^2*b), ( 4, 6 ) = (b), ( 5, 5 ) = (1), ( 4, 1 ) = (a^2), ( 7, 5 ) = (a^2), ( 3, 3 ) = (a^2), ( 6, 5 ) = (a), ( 4, 3 ) = (1), ( 7, 6 ) = (a), ( 1, 3 ) = (a), ( 2, 5 ) = (b), ( 2, 7 ) = (a^2*b), ( 5, 4 ) = (a*b), ( 1, 7 ) = (a^2*b), ( 2, 6 ) = (a*b), ( 7, 2 ) = (a^2*b), ( 5, 7 ) = (a), ( 4, 7 ) = (a*b), ( 6, 4 ) = (a^2*b), ( 2, 3 ) = (a), ( 1, 5 ) = (b), ( 5, 6 ) = (a^2), ( 1, 1 ) = (omicron), ( 6, 1 ) = (a*b), ( 6, 2 ) = (a*b), ( 4, 2 ) = (a^2), ( 1, 2 ) = (1), ( 7, 7 ) = (1), ( 2, 4 ) = (a^2), ( 3, 2 ) = (a), ( 7, 4 ) = (b), ( 6, 7 ) = (a^2), ( 4, 4 ) = (a), ( 5, 3 ) = (a^2*b), ( 5, 2 ) = (b), ( 3, 1 ) = (a), ( 3, 7 ) = (b), ( 5, 1 ) = (b), ( 2, 1 ) = (1), ( 7, 3 ) = (a*b), ( 3, 5 ) = (a*b), ( 1, 6 ) = (a*b) ] )

(17)
> 

GT(x, 4, y, 2, 3)

array( 1 .. 9, 1 .. 9, [( 8, 2 ) = (x^2*y), ( 3, 4 ) = (x^3), ( 2, 2 ) = (1), ( 6, 6 ) = (1), ( 3, 6 ) = (x*y), ( 1, 4 ) = (x^2), ( 6, 3 ) = (x^3*y), ( 4, 5 ) = (x), ( 8, 6 ) = (x^2), ( 7, 1 ) = (x*y), ( 9, 4 ) = (x*y), ( 8, 9 ) = (x^3), ( 4, 6 ) = (x^2*y), ( 1, 9 ) = (x^3*y), ( 5, 5 ) = (x^2), ( 4, 1 ) = (x^2), ( 9, 7 ) = (x^2), ( 7, 5 ) = (x^2*y), ( 3, 3 ) = (x^2), ( 5, 9 ) = (x^2*y), ( 9, 1 ) = (x^3*y), ( 6, 5 ) = (x*y), ( 4, 3 ) = (x^3), ( 9, 8 ) = (x), ( 7, 6 ) = (x), ( 1, 3 ) = (x), ( 2, 5 ) = (x^3), ( 8, 1 ) = (x^2*y), ( 2, 7 ) = (x*y), ( 5, 8 ) = (x*y), ( 6, 8 ) = (x^2), ( 5, 4 ) = (x), ( 9, 6 ) = (x^3), ( 2, 9 ) = (x^3*y), ( 1, 7 ) = (x*y), ( 2, 6 ) = (y), ( 7, 2 ) = (x*y), ( 7, 8 ) = (x^3), ( 5, 7 ) = (y), ( 4, 7 ) = (x^3*y), ( 8, 7 ) = (x), ( 6, 4 ) = (x^2*y), ( 2, 3 ) = (x), ( 8, 8 ) = (1), ( 1, 5 ) = (x^3), ( 5, 6 ) = (x^3*y), ( 1, 1 ) = (omicron), ( 6, 1 ) = (y), ( 7, 9 ) = (x^2), ( 8, 4 ) = (y), ( 6, 2 ) = (y), ( 4, 2 ) = (x^2), ( 9, 3 ) = (x^2*y), ( 9, 2 ) = (x^3*y), ( 1, 2 ) = (1), ( 3, 8 ) = (x^3*y), ( 7, 7 ) = (1), ( 4, 8 ) = (y), ( 2, 4 ) = (x^2), ( 4, 9 ) = (x*y), ( 3, 2 ) = (x), ( 7, 4 ) = (x^3*y), ( 6, 7 ) = (x^3), ( 4, 4 ) = (1), ( 8, 3 ) = (x*y), ( 5, 3 ) = (1), ( 2, 8 ) = (x^2*y), ( 5, 2 ) = (x^3), ( 3, 1 ) = (x), ( 6, 9 ) = (x), ( 3, 7 ) = (x^2*y), ( 5, 1 ) = (x^3), ( 9, 9 ) = (1), ( 2, 1 ) = (1), ( 1, 8 ) = (x^2*y), ( 8, 5 ) = (x^3*y), ( 9, 5 ) = (y), ( 7, 3 ) = (y), ( 3, 9 ) = (y), ( 3, 5 ) = (1), ( 1, 6 ) = (y) ] )

(18)

 

The abelian case is when  r =1.

 

> 

GT(a, 2, b, 2, 1)

array( 1 .. 5, 1 .. 5, [( 3, 4 ) = (a*b), ( 2, 2 ) = (1), ( 1, 4 ) = (b), ( 4, 5 ) = (a), ( 5, 5 ) = (1), ( 4, 1 ) = (b), ( 3, 3 ) = (1), ( 4, 3 ) = (a*b), ( 1, 3 ) = (a), ( 2, 5 ) = (a*b), ( 5, 4 ) = (a), ( 2, 3 ) = (a), ( 1, 5 ) = (a*b), ( 1, 1 ) = (omicron), ( 4, 2 ) = (b), ( 1, 2 ) = (1), ( 2, 4 ) = (b), ( 3, 2 ) = (a), ( 4, 4 ) = (1), ( 5, 3 ) = (b), ( 5, 2 ) = (a*b), ( 3, 1 ) = (a), ( 5, 1 ) = (a*b), ( 2, 1 ) = (1), ( 3, 5 ) = (b) ] )

(19)
> 

``

Power of element  "z^(s ) in a group G(a,m,b,n,r)."

 

> 

PrC:=proc(z,s,m,n,r)

> 

local t,p,q;

> 

p := (z,w)->(Re(z)+r^Im(z)*Re(w)) mod m +I*(Im(z)+Im(w) mod n);

> 

q:=t->p(q(t-1),z);

> 

q(1):=z;

> 

q(s);  

> 

end proc;

> 

 

proc (z, s, m, n, r) local t, p, q; p := proc (z, w) options operator, arrow; `mod`(Re(z)+r^Im(z)*Re(w), m)+I*(`mod`(Im(z)+Im(w), n)) end proc; q := proc (t) options operator, arrow; p(q(t-1), z) end proc; q(1) := z; q(s) end proc

(20)
> 

PrC(2+I, 3, 5, 2, 3);

I

(21)
> 

``

Power of element  "a^(j)*b^(j) in a group G(a,m,b,n,r)."

 

 

> 

NULL

> 

PrG:=proc(i,j,s,m,n,r) local z;

> 

z:=PrC(Complex(i, j), s, m, n, r);a^Re(z)*b^Im(z);

> 

end proc;

proc (i, j, s, m, n, r) local z; z := PrC(Complex(i, j), s, m, n, r); a^Re(z)*b^Im(z) end proc

(22)
> 

PrG(1, 1, 7, 4, 2, 3);

a*b

(23)
> 

``

Inverse Elements Tables  

 

Inverse elements table ITC of GC is as follows.

> 

ITC:=proc(m,n,r)

> 

local M,L,inv,IL;inv := z->Complex(`mod`((m-Re(z))*r^(2-Im(z)), m), `mod`(n-Im(z), n)) ;

> 

M:=Transpose(Matrix(m,n,(i,j)->Complex(i-1,j-1)));

> 

L:= [seq(seq(M(i,j),j=1..m),i=1..n)];

> 

IL:=Map(inv,L);

> 

Transpose(Matrix(2, m*n, [L, IL]));

> 

end proc;

> 

 

proc (m, n, r) local M, L, inv, IL; inv := proc (z) options operator, arrow; Complex(`mod`((m-Re(z))*r^(2-Im(z)), m), `mod`(n-Im(z), n)) end proc; M := LinearAlgebra:-Transpose(Matrix(m, n, proc (i, j) options operator, arrow; Complex(i-1, j-1) end proc)); L := [seq(seq(M(i, j), j = 1 .. m), i = 1 .. n)]; IL := LinearAlgebra:-Map(inv, L); LinearAlgebra:-Transpose(Matrix(2, m*n, [L, IL])) end proc

(24)
> 

 

> 

ITC(5, 2, 4)

Matrix([[0, 0], [1, 4], [2, 3], [3, 2], [4, 1], [I, I], [1+I, 1+I], [2+I, 2+I], [3+I, 3+I], [4+I, 4+I]])

(25)
> 

 

Inverse elements table ITG of G is as follows.

 

> 

ITG:=proc(a,m,b,n,r)

> 

  local hh; hh:=w->a^Re(w)*b^Im(w);

> 

  Map(hh,ITC(m,n,r));

> 

end proc;

proc (a, m, b, n, r) local hh; hh := proc (w) options operator, arrow; a^Re(w)*b^Im(w) end proc; LinearAlgebra:-Map(hh, ITC(m, n, r)) end proc

(26)
> 

NULL

> 

ITG(a, 5, b, 2, 4)

Matrix([[1, 1], [a, a^4], [a^2, a^3], [a^3, a^2], [a^4, a], [b, b], [a*b, a*b], [a^2*b, a^2*b], [a^3*b, a^3*b], [a^4*b, a^4*b]])

(27)
> 

``

Order Tables

The order OrdC of an element z in GC and order table OTC of GC

 

> 

OrdC:=proc(z,m,n,r)

> 

local t,p,q;

> 

z;p := (z,w)->(Re(z)+r^Im(z)*Re(w)) mod m +I*(Im(z)+Im(w) mod n);

> 

q:=n->p(q(n-1),z);

> 

q(1):=z;

> 

for t from 1 to m*n do if is(q(t)=0) then return(t) fi;od;   

> 

end proc;

proc (z, m, n, r) local t, p, q; z; p := proc (z, w) options operator, arrow; `mod`(Re(z)+r^Im(z)*Re(w), m)+I*(`mod`(Im(z)+Im(w), n)) end proc; q := proc (n) options operator, arrow; p(q(n-1), z) end proc; q(1) := z; for t to m*n do if is(q(t) = 0) then return t end if end do end proc

(28)
> 

``

> 

``

> 

OrdC(1, 4, 2, 3);

4

(29)
> 

OrdC(1+I, 4, 2, 3);

2

(30)
> 

 

> 

OTC:=proc(m,n,r)

> 

local L,OL,ord1;

> 

L:= GC(a,m,b,n,r);

> 

ord1:=z->OrdC(z,m,n,r);

> 

OL:=Map(ord1,L);

> 

Transpose(Matrix(2, m*n, [L, OL]));

> 

end proc;

> 

 

> 

 

proc (m, n, r) local L, OL, ord1; L := GC(a, m, b, n, r); ord1 := proc (z) options operator, arrow; OrdC(z, m, n, r) end proc; OL := LinearAlgebra:-Map(ord1, L); LinearAlgebra:-Transpose(Matrix(2, m*n, [L, OL])) end proc

(31)
> 

NULL

> 

OTC(5, 2, 4)

Matrix([[0, 1], [1, 5], [2, 5], [3, 5], [4, 5], [I, 2], [1+I, 2], [2+I, 2], [3+I, 2], [4+I, 2]])

(32)
> 

``

The order of element in G and the order table OTG of G

 

> 

OrdG:=proc(i,j,a,m,b,n,r) local z;

> 

z:=Complex(i,j);

> 

OrdC(z,m,n,r);

> 

end proc;

proc (i, j, a, m, b, n, r) local z; z := Complex(i, j); OrdC(z, m, n, r) end proc

(33)
> 

``

> 

OrdG(3, 0, a, 4, b, 2, 3);

 

4

(34)
> 

``

> 

OTG:=proc(a,m,b,n,r) local A,L,K,hh,LL;

> 

A:=OTC(m,n,r);

> 

L:=convert(col(A,1),list);

> 

K:=convert(col(A,2),list);

> 

hh:=w->a^Re(w)*b^Im(w);

> 

LL:=Map(hh,L);

> 

transpose(Matrix(2,m*n,[LL,K]));

> 

end proc;

proc (a, m, b, n, r) local A, L, K, hh, LL; A := OTC(m, n, r); L := convert(linalg:-col(A, 1), list); K := convert(linalg:-col(A, 2), list); hh := proc (w) options operator, arrow; a^Re(w)*b^Im(w) end proc; LL := LinearAlgebra:-Map(hh, L); linalg:-transpose(Matrix(2, m*n, [LL, K])) end proc

(35)
> 

``

> 

OTG(x, 5, y, 2, 4)

array( 1 .. 10, 1 .. 2, [( 3, 2 ) = (5), ( 5, 1 ) = (x^4), ( 8, 2 ) = (2), ( 10, 2 ) = (2), ( 5, 2 ) = (5), ( 8, 1 ) = (x^2*y), ( 2, 1 ) = (x), ( 9, 2 ) = (2), ( 1, 2 ) = (1), ( 2, 2 ) = (5), ( 1, 1 ) = (1), ( 10, 1 ) = (x^4*y), ( 7, 2 ) = (2), ( 9, 1 ) = (x^3*y), ( 6, 1 ) = (y), ( 3, 1 ) = (x^2), ( 4, 2 ) = (5), ( 6, 2 ) = (2), ( 4, 1 ) = (x^3), ( 7, 1 ) = (x*y) ] )

(36)
> 

``

 Centerlizer and Center

 Centeralizer CentC in GC and centeralizer CentG in G as well as CenterC and CenterG are found below  

> 

NULL

> 

NULL

> 

CentC:=proc(u,a,m,b,n,r)

> 

   local E,p;

> 

   E:=GC(a,m,b,n,r);

> 

   p := (z,w)->(Re(z)+r^Im(z)*Re(w)) mod m +I*(Im(z)+Im(w) mod n);  

> 

  { select(z->is(p(u,z)=p(z,u)),E)[]};

> 

end proc;

proc (u, a, m, b, n, r) local E, p; E := GC(a, m, b, n, r); p := proc (z, w) options operator, arrow; `mod`(Re(z)+r^Im(z)*Re(w), m)+I*(`mod`(Im(z)+Im(w), n)) end proc; {select(proc (z) options operator, arrow; is(p(u, z) = p(z, u)) end proc, E)[]} end proc

(37)
> 

NULL

> 

CentC(1+I, a, 3, b, 2, 3)

{0, 1+I}

(38)
> 

``

> 

CentG:=proc(i,j,a,m,b,n,r) local S,hh; hh:=w->a^Re(w)*b^Im(w);

> 

S:=CentC(Complex(i,j),a,m,b,n,r);

> 

Map(hh,S);

> 

end proc;

> 

 

> 

 

proc (i, j, a, m, b, n, r) local S, hh; hh := proc (w) options operator, arrow; a^Re(w)*b^Im(w) end proc; S := CentC(Complex(i, j), a, m, b, n, r); LinearAlgebra:-Map(hh, S) end proc

(39)
> 

CentG(1, 1, a, 3, b, 2, 2)

{1, a*b}

(40)
> 

CentG(2, 1, a, 4, b, 2, 3)

{1, b, a^2, a^2*b}

(41)
> 

NULL

> 

CenterC:=proc(a,m,b,n,r)

> 

local S,E;E:=GC(a,m,b,n,r);

> 

S := seq(CentC(z, a, m, b, n, r), `in`(z, E));

> 

`intersect`(S);

> 

end proc;

proc (a, m, b, n, r) local S, E; E := GC(a, m, b, n, r); S := seq(CentC(z, a, m, b, n, r), `in`(z, E)); `intersect`(S) end proc

(42)
> 

``

> 

``

> 

CenterC(a, 6, b, 2, 5);

{0, 3}

(43)
> 

CenterG:=proc(a,m,b,n,r)

> 

  local L,hh;

> 

  L:=CenterC(a,m,b,n,r);

> 

hh:=w->a^Re(w)*b^Im(w);

> 

Map(hh,L);

> 

end proc;

proc (a, m, b, n, r) local L, hh; L := CenterC(a, m, b, n, r); hh := proc (w) options operator, arrow; a^Re(w)*b^Im(w) end proc; LinearAlgebra:-Map(hh, L) end proc

(44)
> 

CenterG(a, 3, b, 4, 2);

{1, b^2}

(45)
> 

CenterG(a, 6, b, 2, 5);

{1, a^3}

(46)
> 

CenterG(x, 7, y, 2, 6);

{1}

(47)
> 

CenterG(a, 8, b, 2, 7);

{1, a^4}

(48)
> 

``

Conjugate Classes

ConjC and ConjG are to find cojugate class of an element in GC and G respectively.

ConjClassesC and ConjClassesG are to find cojugate classes of GC and G respectively.

 

> 

``

> 

ConjC:=proc(z,a,m,b,n,r)

> 

 local p,inv,E;  E:=GC(a,m,b,n,r);

> 

 p := (z,w)->(Re(z)+r^Im(z)*Re(w)) mod m +I*(Im(z)+Im(w) mod n);  

> 

 inv := z->Complex(`mod`((m-Re(z))*r^(2-Im(z)), m), `mod`(n-Im(z), n)) ;

> 

 {seq(p(p(w,z),inv(w)),w in E)};

> 

end proc;

> 

 

proc (z, a, m, b, n, r) local p, inv, E; E := GC(a, m, b, n, r); p := proc (z, w) options operator, arrow; `mod`(Re(z)+r^Im(z)*Re(w), m)+I*(`mod`(Im(z)+Im(w), n)) end proc; inv := proc (z) options operator, arrow; Complex(`mod`((m-Re(z))*r^(2-Im(z)), m), `mod`(n-Im(z), n)) end proc; {seq(p(p(w, z), inv(w)), `in`(w, E))} end proc

(49)
> 

NULL

> 

NULL

> 

ConjC(1+I, a, 4, b, 2, 3);

{1+I, 3+I}

(50)
> 

ConjC(0, a, 4, b, 2, 3);

{0}

(51)
> 

ConjC(2, a, 4, b, 2, 3);

{2}

(52)
> 

ConjG:=proc(i,j,a,m,b,n,r)

> 

local z,hh,S;

> 

z:=Complex(i,j);S:=ConjC(z,a,m,b,n,r);

> 

hh:=w->a^Re(w)*b^Im(w);

> 

Map(hh,S);

> 

end proc;

> 

 

proc (i, j, a, m, b, n, r) local z, hh, S; z := Complex(i, j); S := ConjC(z, a, m, b, n, r); hh := proc (w) options operator, arrow; a^Re(w)*b^Im(w) end proc; LinearAlgebra:-Map(hh, S) end proc

(53)
> 

NULL

> 

ConjG(1, 1, a, 4, b, 2, 3);

{a*b, a^3*b}

(54)
> 

``

> 

ConjClassesC:=proc(a,m,b,n,r)

> 

local E,con1; con1:=z->ConjC(z,a,m,b,n,r);

> 

E:=GC(a,m,b,n,r);

> 

{seq(con1(z),z in E)};

> 

end proc;

proc (a, m, b, n, r) local E, con1; con1 := proc (z) options operator, arrow; ConjC(z, a, m, b, n, r) end proc; E := GC(a, m, b, n, r); {seq(con1(z), `in`(z, E))} end proc

(55)
> 

``

> 

``

> 

ConjClassesC(a, 4, b, 2, 3);

{{0}, {2}, {1, 3}, {I, 2+I}, {1+I, 3+I}}

(56)
> 

``

> 

ConjClassesG:=proc(a,m,b,n,r)

> 

local H,hh;

> 

hh:=w->a^Re(w)*b^Im(w);

> 

H:=ConjClassesC(a,m,b,n,r);seq(Map(hh,X),X=H);

> 

end proc;

> 

 

proc (a, m, b, n, r) local H, hh; hh := proc (w) options operator, arrow; a^Re(w)*b^Im(w) end proc; H := ConjClassesC(a, m, b, n, r); seq(LinearAlgebra:-Map(hh, X), X = H) end proc

(57)
> 

NULL

> 

ConjClassesG(a, 4, b, 2, 3);

{1}, {a^2}, {a, a^3}, {b, a^2*b}, {a*b, a^3*b}

(58)
> 

ConjClassesG(a, 3, b, 2, 2);

{1}, {a, a^2}, {b, a*b, a^2*b}

(59)
> 

ConjClassesG(a, 3, b, 4, 2);

{1}, {b^2}, {a, a^2}, {a*b^2, a^2*b^2}, {b, a*b, a^2*b}, {b^3, a*b^3, a^2*b^3}

(60)

 

Remark

It is known that a subgroup and a quotient group of  a metacyclic group are metacyclic. One may use the same treatment to find all subgroups of a metacyclic group.

> 

``