8.24 RepresentativesOperation

RepresentativesOperation( G, d )
RepresentativesOperation( G, d, operation )

RepresentativesOperation returns a list of representatives of the points in the orbit of the point d under the group G.

The ordering of the representatives corresponds to the ordering of the points in the orbit as returned by Orbit (see Orbit). Therefore List( RepresentativesOperation(G,d), r-d^r ) = Orbit(G,d).

An element g of G is called a representative for the point e in the orbit of d under G if g maps d to e, i.e., d^g = e. Note that the set of such representatives that map d to e forms a right coset of the stabilizer of d in G (see Stabilizer). The set of all representatives of the orbit of d under G thus forms a system of representatives of the right cosets of the stabilizer of d in G.

RepresentativesOperation accepts a function operation of two arguments d and g as optional third argument, which specifies how the elements of G operate (see Other Operations).

    gap> g := Group( (1,2,3)(6,7), (3,4,5)(7,8) );;
    gap> RepresentativesOperation( g, 1 );
    [ (), (1,2,3)(6,7), (1,3,2), (1,4,5,3,2)(7,8), (1,5,4,3,2) ]
    gap> Orbit( g, [1,2], OnSets );
    [ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 2, 4 ], [ 1, 4 ], [ 3, 4 ],
      [ 2, 5 ], [ 1, 5 ], [ 4, 5 ], [ 3, 5 ] ]
    gap> RepresentativesOperation( g, [1,2], OnSets );
    [ (), (1,2,3)(6,7), (1,3,2), (1,2,4,5,3)(6,8,7), (1,4,5,3,2)(7,8),
      (1,3,2,4,5)(6,8), (1,2,5,4,3)(6,7), (1,5,4,3,2), (1,4,3,2,5)(6,7,8),
      (1,3,2,5,4) ] 

RepresentativesOperation calls
G.operations.RepresentativesOperation( G, d, operation )
and returns the value. Note that the third argument is not optional for functions called this way.

The default function called this way is GroupOps.RepresentativesOperation, which computes the orbit of d with the normal algorithm, but remembers for each point e in the orbit a representative r_e. When a generator g of G takes an old point e to a point f not yet in the orbit, the representative r_f for f is computed as r_e g. Special categories of groups overlay this default function with more efficient functions.

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