To save a record of your Gap session enter something like
gap> LogTo("filename"); 
or
gap> LogTo("may10.07a");

In fact I did this on Gap just before the first line you can read above.
Naturally, when done I looked at this file and edited it for appearance, 
removed Gap error messages, the Gap prompts before the current line, etc.

There are a couple of ways to write your own Gap functions and programmes.
The quickest way, especially for simple functions, is to directly enter
the command. For example, if you want Gap to take  as input a permutation p
and give as output p, p^2  ... p^5 enter this:

gap> pow:=p->[p,p^2,p^3,p^4,p^5];
function( p ) ... end

gap> pow( (1,2,3,4,5) );
[ (1,2,3,4,5), (1,3,5,2,4), (1,4,2,5,3), (1,5,4,3,2), () ]
gap> pow( (1,2)*(2,3)*(3,4) );

[ (1,4,3,2), (1,3)(2,4), (1,2,3,4), (), (1,4,3,2) ]


However, for editing, debugging and future use, it is often better
to independently create the function in a separate file, say   wombat,
then read it in usuing the Gap command Read("wombat"); note that file
names inside Gap come between double quotes.

That way, if there is a goof, you can edit and Read the file again, etc.
For eample, suppose in your text editor you compose a file called   wombat
with the following lines:

newpow:= function(p) local j; # stuff after a # is ignored
 for j in [1..5] do
   Print("power j = ",j," gives ", p^j,"\n","Have nice day!","\n");
                od;
 end; # end of this silly function



Then the command
gap> Read("wombat");


silently imports the function newpow for your use:

gap> newpow( (1,2)*(2,3)*(3,4) );
power j = 1 gives (1,4,3,2)
Have nice day!
power j = 2 gives (1,3)(2,4)
Have nice day!
power j = 3 gives (1,2,3,4)
Have nice day!
power j = 4 gives ()
Have nice day!
power j = 5 gives (1,4,3,2)
Have nice day!


Exercise 1: Write a Gap function   ref    with input m which  produces the 
2 x 2  orthoogonal matrix for reflection in the line y = mx.


Remark: Gap can't do general sines and cosines, so we could
take the following  approach for rotations.

Exercise 2: Assuming ref is available in the background, write a function
rot   of (m,n) which produces the rotation matrix for the product of reflections
in lines of slope m and n.

Regarding trigonometric functions we can actually do something, since

	exp(2*pi*i/n) = cos(2*pi/n) + i sin(2*pi*n)
is an important `algebraic number' and hence comes built into Gap.
The penalty is that simple rela numbers might end up expressed as 
messy combinations of complex numbers.


Exercise 3: Find out about Gaps way of dealing with cyclotomic field 
involving the complex number z := exp(2*pi*i/n) = cos(2*pi/n) + i sin(2*pi*n).

Now given  an input  rational number r, find out how to get the numerator
and denominator of r.
Next write   a function rotrat of r which
gives the 2 x 2 orthogonal rotation matrix for an angle of r degrees.


		***************************************
We can represent the line  ax + by = c  in the PLANE using the 
row vector [a,b,-c] and the point (x,y) by the column with entries x,y,1;

Actually in Gap this would be [[a,b,-c]] and [[x],[y],[1]], as in
gap> line:=[[2,3,-4]]; A:=[[11],[-6],[1]];B:=[[10],[-6],[1]];;
[ [ 2, 3, -4 ] ]
[ [ 11 ], [ -6 ], [ 1 ] ]
gap> Display(line);Display(A);Display(B);
[ [   2,   3,  -4 ] ]
[ [  11 ],
  [  -6 ],
  [   1 ] ]
[ [  10 ],
  [  -6 ],
  [   1 ] ]
gap> line*A;line*B;
[ [ 0 ] ]
[ [ -2 ] ]


Thus A lies on the line but B doesn't.

Exercise 4. Write a function    glide   whose input is a 3 x 3 matrix L  
whose rows are the line vectors for three lines, something like
             [ a1, b1, -c1]
       L  =  [ a2, b2, -c2]   (this isn't quite correct Gap syntax).
             [ a3, b3, -c3]
             
In actual use, the input will be a concrete matrix like

gap> L:=[[1,2,3],[4,5,6],[9,8,7]];;

(So the third line here has equation 9x + 8y = -7)
                            
The output should be a   list of three things:
	first: a 3 by 2 matrix  whose two columns are the initial point A
	and terminal point B for the glide step AB (this means the axis
	is line AB and the parallel shift is vector AB). Note that the 
	bottom row of this matrix  should be [[1,1]], according to how we 
	represent points in the plane by three dimensional columns.
	
	second: the row vector for the equation of the axis line
	
	third: the length of the glide step (same as length of segment
	AB).

Remark: input data will typically be rational numbers; output three 
may require roots;
so you will have to be sure Gap is happy to take square roots. This should work
but may need a bit of fussing.

Enhancement: give a separate output indication when the glide degenerates to an
ordinary reflection; and in this case print the line vector for the mirror,
If you want to be really fancy, give text output, as in the 
  
       The glide has axis  3x + 5y = 7 and step length root(12).