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% Solutions to above sample exercises
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\choosett{apl}

\noindent
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\noindent
As the index of the neutral element we use the index origin \BX@IO@ which
usually has the value @0@. Then  $S(N)=
\{0,\dots,N-1\}$, given by the vector \IO@N@.
An example on groups are the cyclic groups $({\bf Z}_n,+)$
the group tables of which are generated by the \APL\ function @ZNPLUS@:

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL Z_ZNPLUS N;@BXIO
[1]   @BXIO_0
[2]   Z_N@AB(@ION)@SO.+@ION
    @DL
\endtt
}\smallskip

\item{1.}  The matrices represent binary operations of $S(N)$,
           since they are $N\times N$-matrices with elements from
           $S(N)$. They are all associative and also commutative except for
           the case (b). This can be seen by the function @TEST@:

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL Z_TEST B
[1]  " B IS A BINARY OPERATION. THE FUNCTION RETURNS A BOOLEAN 2-VECTOR
[2]  " (B ASSOCIATIVE, B COMMUTATIVE)
[3]   Z_(&/&/&/B[B;]=B[;B]),&/&/B=@TRB
    @DL
\endtt
}\smallskip

\item{2.}

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL P_X GPOWER N;I
[1]  " G GLOBAL
[2]   P_@BXIO @DM I_0
[3]  TEST:@GO(N<I_I+1)/0
[4]   P_G[P;X]
[5]   @GOTEST
    @DL
\endtt
}\smallskip

\item{3.}

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL P_X BGPOWER N;IJ
[1]  " G GLOBAL
[2]   P_@BXIO
[3]  NEXTJ:@GO(0=N,IJ_2@ABN)/0,SQX
[4]   P_G[P;X]
[5]  SQX:X_G[X;X]
[6]   N_(N-IJ)%2
[7]   @GONEXTJ
    @DL
\endtt
}

\item{}  A comment: if $i_j=0$, then the power is not increased,
         but the square $x^{2^{j+1}}=(x^{2^j})^2$ is computed.
         The number of iterations is $k$; $n = i_0+i_12+\cdots+i_k2^k \ge 2^k$,
         when $i_k \not= 0$, and hence $k \le \log_2(n)$.
         Thus, the complexity is $O(\log_2(n))$.
\smallskip

\vfill\eject
\item{4.}

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL Z_A GTSGP G
[1]  " RETURNS THE SUBGROUP OF G GENERATED BY A
[2]   Z_,A
[3]  TEST:@GO(&/&/G[Z;Z]@EPZ)/FOUND
[4]   Z_Z UNION G[Z;Z]
[5]   @GOTEST
[6]  FOUND:Z_Z[@GUZ]
    @DL
\endtt
}

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL Z_A UNION B;V;@BXIO
[1]   V_(,A),,B
[2]   @BXIO_1
[3]   Z_,CLEAN((@ROV),1)@ROV
    @DL
\endtt
}

The auxiliary function @CLEAN@ was given earlier.
\bigskip

\item{5.}

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL Z_INV G
[1]  " RETURNS THE VECTOR OF INVERSE ELEMENTS OF G
[2]   (@BXIO=,G)/,(@ROG)@ROG[@BXIO;]
    @DL
\endtt
}\smallskip

\item{6.}

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL H_A BGTSGP G;Y
[1]  " RETURNS THE SUBGROUP OF G GENERATED BY A
[2]   H_Y_@BXIO
[3]  B:@GO(0=@ROY_(,G[Y;A])MINUS H)/0
[4]   H_H UNION Y
[5]   @GOB
    @DL
\endtt
}

\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL Z_A MINUS B
[1]   Z_(@NTA@EPB)/A
    @DL
\endtt
}\smallskip

\item{7.}  If the elements of $G_i$ have been indexed by the interval
           $[0,n_i-1]$, the elements of $G_1\times G_2$ become indexed
           in a natural way by the elements of the Cartesian product
           $[0,n_1-1]\times[0,n_2-1]$. With the bijection
           $(i,j) \mapsto in_2+j:[0,n_1-1]\times[0,n_2-1]
           \longrightarrow[0,n_1n_2-1]$
           (the inverse $k\mapsto((k-(k \bmod n_2))/n_2,k \bmod n_2)$
           selects the quotient and remainder in the division by $n_2$)
           we get $[0,n_1n_2-1]$ as the index set.

\vfill\eject
\hskip\parskip\vbox{\hsize=15truecm
\begintt
    @DL G_G1 PROD G2;@BXIO;I;J;IREM;JREM;N1;N2;N
[1]   N_(N1_(@ROG1)[1])#N2_(@ROG2)[1] @DM I_@BXIO_0
[2]   G_(N,N)@RO0
[3]  JLOOP:J_0
[4]  CORE:G[I;J]_(G1[(I-IREM)%N2;(J-JREM)%N2]#N2)+G2[IREM_N2@ABI;JREM_N2@ABJ]
[5]   @GO(N>J_J+1)/CORE
[6]   @GO(N>I_I+1)/JLOOP
    @DL
\endtt
}

Example:

\hskip\parskip\vbox{\hsize=15truecm
\begintt
      (ZNPLUS 2) PROD ZNPLUS 10
 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19
 1  2  3  4  5  6  7  8  9  0 11 12 13 14 15 16 17 18 19 10
 2  3  4  5  6  7  8  9  0  1 12 13 14 15 16 17 18 19 10 11
 3  4  5  6  7  8  9  0  1  2 13 14 15 16 17 18 19 10 11 12
 4  5  6  7  8  9  0  1  2  3 14 15 16 17 18 19 10 11 12 13
 5  6  7  8  9  0  1  2  3  4 15 16 17 18 19 10 11 12 13 14
 6  7  8  9  0  1  2  3  4  5 16 17 18 19 10 11 12 13 14 15
 7  8  9  0  1  2  3  4  5  6 17 18 19 10 11 12 13 14 15 16
 8  9  0  1  2  3  4  5  6  7 18 19 10 11 12 13 14 15 16 17
 9  0  1  2  3  4  5  6  7  8 19 10 11 12 13 14 15 16 17 18
10 11 12 13 14 15 16 17 18 19  0  1  2  3  4  5  6  7  8  9
11 12 13 14 15 16 17 18 19 10  1  2  3  4  5  6  7  8  9  0
12 13 14 15 16 17 18 19 10 11  2  3  4  5  6  7  8  9  0  1
13 14 15 16 17 18 19 10 11 12  3  4  5  6  7  8  9  0  1  2
14 15 16 17 18 19 10 11 12 13  4  5  6  7  8  9  0  1  2  3
15 16 17 18 19 10 11 12 13 14  5  6  7  8  9  0  1  2  3  4
16 17 18 19 10 11 12 13 14 15  6  7  8  9  0  1  2  3  4  5
17 18 19 10 11 12 13 14 15 16  7  8  9  0  1  2  3  4  5  6
18 19 10 11 12 13 14 15 16 17  8  9  0  1  2  3  4  5  6  7
19 10 11 12 13 14 15 16 17 18  9  0  1  2  3  4  5  6  7  8
\endtt
}

\end