Advanced Programming Guide

136 • Chapter 3: Programming with Modules> local i;> [ seq( b[ i ], i = a ) ]> end proc;> G:-mul := () -> foldl( G:-‘.‘, G:-id, args );> G:-inv := proc( g )> local i, a;> a := array( 1 .. deg );> for i from 1 to deg do> a[ g[ i ] ] := i> end do;> [ seq( a[ i ], i = 1 .. deg ) ]> end proc;> G:-member := proc( g, S, pos::name )> if nargs = 1 then> type( g, ’list( posint )’ )> and { op( g ) } = { $ 1 .. deg }> else> :-member( args )> end if> end proc;> G:-eq := ( a, b ) -> evalb( a = b );> G:-gens := gens;> eval( G, 1 )> end proc:For example, to construct the group 〈(12), (123)〉 in the symmetric groupS 4 , use the PermutationGroup constructor as follows.> G := PermutationGroup( 4, { [2,1,3,4], [2,3,1,4] } );G := module()exportid , ‘.‘, mul , inv, eq, member, gens, order, elements;option record ;end moduleTo compute with its elements, use the methods exported by the instantiatedgroup G.> use G in> inv( [ 2,1,3,4 ] ) . [2,3,1,4];> end use;[3, 2, 1, 4]It is useful to provide more specialized permutation group constructorsfor special kinds of groups. Using the general constructorPermutationGroup, and overriding some of the exported methods, youcan define several of these specialized constructors as follows.