Clear();
Print ("Beginning demo program: test_2_4.txt\n\n");
Print ("This demonstrates the use of the following procedures:\n");
Print ("   SubgroupIncidenceMatrix( order, group_id ) \n");
Print ("   MaxChains( mat, allchains ) \n");
Print ("   MaxSubgroupChains( mat, subgroup ) \n\n");
Sleep(5);


# let's create some chains of subgroups
# Begin with the dihedral group on 8 elements

mat := SubgroupIncidenceMatrix(8, 3);

Print ("   Let's create some chains of subgroups.  Remember that the\n");
Print ("dihedral group on 8 elements is SmallGroup(8,3).  So...\n");
Print ("   mat := SubgroupIncidenceMatrix(8,3);\n\n");
PrintArray (mat);
Print ("\n\n");
Sleep(10);

Print ("  This is a partial ordering showing which normal subgroups\n");
Print ("are within others.  Remember:  Rows are strictly contained in\n");
Print ("columns.  Thus NormalSubgroup 1, which is trivial, is in\n");
Print ("every normal subgroup except itself.\n\n\n");
Sleep(10);  


chains := MaxChains( mat );

Print ("So what are the longest chains of subgroups?\n");
Print ("   chains := MaxChains( mat );\n\n");
Print (chains, "\n\n");
Sleep(8);


# so the chain [1,2,3,6] is saying:
# n[1] < n[2] < n[3] < n[6]

g   := SmallGroup(8,3);
n := NormalSubgroups(g);

Print ("   g := SmallGroup(8,3);;\n");
Print ("   n := NormalSubgroups( group );;\n\n");
Print ("So the chain [1,2,3,6] is saying:\n");
Print ("   n[1] < n[2] < n[3] < n[6] \n\n");
Sleep(5);

Print ("n[1] < n[2]?\n", IsSubgroup(n[2], n[1]), "\n");
Print ("n[2] < n[3]?\n", IsSubgroup(n[3], n[2]), "\n");
Print ("n[3] < n[6]?\n", IsSubgroup(n[6], n[3]), "\n\n");
Sleep(8);


# And let's find out what chains have the derived subgroup
# in them.

mychains := MaxSubgroupChains( mat, 2 );

Print ("Let's find which maximal chains contain the derived subgroup.\n");
Print ("We happen to know that the derived subgroup is the second\n");
Print ("normal subgroup.\n");
Print ("   mychains := MaxSubgroupChains( mat, 2 );\n\n");
Print (mychains, "\n\n");