# Chapter 4: Software and Calculations
# 
# 
# The package "groups" also allows you to make calculations with linear
# groups over a field F(p). After loading the package, you must first
# choose a prime with the function ChoosePrime . For example	
# 
> ChoosePrime(11);

                                  11

# 
# All your calculations will then be modulo this prime. You can change
# the prime at any time, but if you do not make a choice none of the
# functions will evaluate. Matrices are usually defined in Maple with
# the function matrix. In this package they are represented as lists of
# lists wrapped with L[] in order to give a mechanism to reduce modulo
# the chosen prime at each step in the calculations. For example, the
# general  2x2  matrix
# 
# 
# is represented as
# 
# L ( [[a,b], [c,d]] )
# 
# The result of the input is then
# 
# [[[a,b], [c,d]], p]
# For example
# 
> a := L ( [[1,2], [3,4]]);

                     a := [[[1, 2], [3, 4]], 11]

# 
# To see the result in matrix notation just enter
# 
> matrix( 2,2, a[1] );

                               [1    2]
                               [      ]
                               [3    4]

# 
# The same functions you used with permutation groups are defined for
# linear groups. For example
# 
> Inverse(a);

                        [[[9, 1], [7, 5]], 11]

# 
# If
# 
> b := L( [[6,3], [7,5]] );

                     b := [[[6, 3], [7, 5]], 11]

# 
# then
# 
> a&*b;

                        [[[9, 2], [2, 7]], 11]

# 
# Now suppose
# 
> ChoosePrime(7);

                                  7

# 
# and
# 
> G := { L([[1, 1], [1, 2]]), L([[2, 3], [4, 1]]), L([[2, 6], [6, 1]]), 
>     L([[1, 0], [0, 1]]), L([[2, 1], [6, 4]]) };

  G := {[[[1, 0], [0, 1]], 7], [[[1, 1], [1, 2]], 7],

        [[[2, 1], [6, 4]], 7], [[[2, 3], [4, 1]], 7],

        [[[2, 6], [6, 1]], 7]}

# 
# then
# 
> Inverse(G);

  {[[[1, 0], [0, 1]], 7], [[[1, 1], [1, 2]], 7],

        [[[2, 1], [6, 4]], 7], [[[2, 3], [4, 1]], 7],

        [[[2, 6], [6, 1]], 7]}

# 
# So G is closed under taking inverses. Let's check whether it is closed
# under products:
# 
> G&*G;

  {[[[3, 6], [1, 1]], 7], [[[6, 4], [3, 5]], 7],

        [[[2, 2], [5, 6]], 7], [[[2, 3], [3, 5]], 7],

        [[[0, 5], [2, 5]], 7], [[[1, 0], [0, 1]], 7],

        [[[3, 4], [3, 0]], 7], [[[5, 4], [4, 2]], 7],

        [[[1, 1], [0, 4]], 7], [[[1, 1], [1, 2]], 7],

        [[[3, 6], [1, 5]], 7], [[[2, 1], [6, 4]], 7],

        [[[2, 3], [4, 1]], 7], [[[5, 5], [4, 3]], 7],

        [[[5, 1], [5, 6]], 7], [[[1, 5], [0, 2]], 7],

        [[[2, 6], [6, 1]], 7]}

# 
# You see immediately that the product of  L( [[1,1], [1,2]] )  with
# itself is not in G. So G is not a linear group.
# 
# 	We can verify that  GL( 2, F(5) )  is generated by the three matrices
# above. First we change the chosen prime to 5.
# 
> ChoosePrime(5);

                                  5

# 
# Let
# 
> H := Group( L([[2, 0], [0, 1]]), L([[0, 1], [1, 0]]), L([[1, 1], [0,
> 1]])):
# 
# Now we check:
# 
> Ord(H);

                                 480

# 
# This is indeed  4^2· 5 ·6 . Can you find three matrices which generate
# GL( 2, F(7) ) ?  GL( 2, F(11) ) ?