# 72.1 Operation for Finitely Presented Algebras

`Operation( F, Q )`

This is the default application of VE. Finitely Presented Algebras), Q is a quotient of a free F-module, and the result is a matrix algebra representing a faithful action on Q.

If Q is the zero module then the matrices have dimension zero, so the result is a null algebra (see NullAlgebra) consisting only of a zero element.

The algebra homomorphism, the isomorphic module for the matrix algebra, and the module homomorphism can be constructed as described in chapters Algebras and Modules.

```    gap> a:= FreeAlgebra( GF(2), 2 );
UnitalAlgebra( GF(2), [ a.1, a.2 ] )
gap> a:= a / [ a.1^2 - a.one, # group algebra of \$V_4\$ over \$GF(2)\$
>              a.2^2 - a.one,
>              a.1*a.2 - a.2*a.1 ];
UnitalAlgebra( GF(2), [ a.1, a.2 ] )
gap> op:= Operation( a, a^1 );
UnitalAlgebra( GF(2),
[ [ [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2),
Z(2)^0 ], [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ],
[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ] ] ] )
gap> Size( op );
16 ```

GAP 3.4.4
April 1997