Orbit( G, d )
Orbit( G, d, operation )
Orbit returns the orbit of the point d, which may be an object of
arbitrary type, under the group G as a list of points.
The points e in the orbit of d under the group G are those points for which a group element g of G exists such that d^g = e.
Suppose G has n generators. First we order the words of the free
monoid with n abstract generators according to length and for words
with equal length lexicographically. So if G has two generators called
a and b the ordering is identity, a, b, a^2, ab, ba, b^2, a^3, ....
Next we order the elements of G that can be written as a product of the
generators, i.e., without inverses of the generators, according to the
first occurrence of a word representing the element in the above
ordering. Then the ordering of points in the orbit returned by
is according to the order of the first representative of each point e,
i.e., the smallest g such that d^g = e. Note that because the orbit
is finite there is for every point in the orbit at least one
representative that can be written as a product in the generators of G.
Orbit 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> Orbit( g, 1 ); [ 1, 2, 3, 4, 5 ] gap> Orbit( g, 2 ); [ 2, 3, 1, 4, 5 ] gap> Orbit( g, [1,6], OnPairs ); [ [ 1, 6 ], [ 2, 7 ], [ 3, 6 ], [ 2, 8 ], [ 1, 7 ], [ 4, 6 ], [ 3, 8 ], [ 2, 6 ], [ 1, 8 ], [ 4, 7 ], [ 5, 6 ], [ 3, 7 ], [ 5, 8 ], [ 5, 7 ], [ 4, 8 ] ]
G.operations.Orbit( 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.Orbit, which performs
an ordinary orbit algorithm. Special categories of groups overlay this
default function with more efficient functions.
Previous Up Top Next