1.22 About Matrix Groups

This section intends to show you the things you could do with matrix groups in GAP. In principle all the set theoretic functions mentioned in chapter Domains and all group functions mentioned in chapter Groups can be applied to matrix groups. However, you should note that at present only very few functions can work efficiently with matrix groups. Especially infinite matrix groups (over the rationals or cyclotomic fields) can not be dealt with at all.

Matrix groups are created in the same way as the other types of groups, by using the function Group. Of course, in this case the arguments have to be invertable matrices over a field.

    gap> m1 := [ [ Z(3)^0, Z(3)^0, Z(3) ],
    >             [ Z(3), 0*Z(3), Z(3) ],
    >             [ 0*Z(3), Z(3), 0*Z(3) ] ];;
    gap> m2 := [ [ Z(3), Z(3), Z(3)^0 ],
    >            [ Z(3), 0*Z(3), Z(3) ],
    >            [ Z(3)^0, 0*Z(3), Z(3) ] ];;
    gap> m := Group( m1, m2 );
    Group( [ [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), 0*Z(3), Z(3) ],
      [ 0*Z(3), Z(3), 0*Z(3) ] ],
    [ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ],
      [ Z(3)^0, 0*Z(3), Z(3) ] ] )

As usual for groups, the matrix group that we have constructed is represented by a record with several entries. For matrix groups, there is one additional entry which holds the field over which the matrix group is written.

    gap> m.field;
    GF(3)

Note that you do not specify the field when you construct the group. Group automatically takes the smallest field over which all its arguments can be written.

At this point there is the question what special functions are available for matrix groups. The size of our group, for example, may be computed using the function Size.

    gap> Size( m );
    864

If we now compute the size of the corresponding general linear group

    gap> (3^3 - 3^0) * (3^3 - 3^1) * (3^3 - 3^2);
    11232

we see that we have constructed a proper subgroup of index 13 of GL(3,3).

Let us now set up a subgroup of m, which is generated by the matrix m2.

    gap> n := Subgroup( m, [ m2 ] );
    Subgroup( Group( [ [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), 0*Z(3), Z(3) ],
      [ 0*Z(3), Z(3), 0*Z(3) ] ],
    [ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ],
      [ Z(3)^0, 0*Z(3), Z(3) ] ] ),
    [ [ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ],
          [ Z(3)^0, 0*Z(3), Z(3) ] ] ] )
    gap> Size( n );
    6

And to round up this example we now compute the centralizer of this subgroup in m.

    gap> c := Centralizer( m, n );
    Subgroup( Group( [ [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), 0*Z(3), Z(3) ],
      [ 0*Z(3), Z(3), 0*Z(3) ] ],
    [ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ],
      [ Z(3)^0, 0*Z(3), Z(3) ] ] ),
    [ [ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ],
          [ Z(3)^0, 0*Z(3), Z(3) ] ],
      [ [ Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3), 0*Z(3) ],
          [ 0*Z(3), 0*Z(3), Z(3) ] ] ] )
    gap> Size( c );
    12

In this section you have seen that matrix groups are constructed in the same way that all groups are constructed. You have also been warned that only very few functions can work efficiently with matrix groups. See chapter Matrix Groups to read more about matrix groups.

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