permgrps.spad line 1 [edit on github]
PermutationGroup implements permutation groups acting on a set S, i.e. all subgroups of the symmetric group of S, represented as a list of permutations (generators). Note that therefore the objects are not members of the FriCAS category Group. Using the idea of base and strong generators by Sims, basic routines and algorithms are implemented so that the word problem for permutation groups can be solved.
gp1 < gp2 returns true if and only if gp1 is a proper subgroup of gp2.
gp1 <= gp2 returns true if and only if gp1 is a subgroup of gp2. Note: because of a bug in the parser you have to call this function explicitly by gp1 <=$(PERMGRP S) gp2.
base(gp) returns a base for the group gp.
coerce(ls) coerces a list of permutations ls to the group generated by this list.
coerce(gp) returns the generators of the group gp.
degree(gp) returns the number of points moved by all permutations of the group gp.
elt(gp, i) returns the i-th generator of the group gp.
generators(gp) returns the generators of the group gp.
initializeGroupForWordProblem(gp) initializes the group gp for the word problem. Notes: it calls the other function of this name with parameters 0 and 1: initializeGroupForWordProblem(gp, 0, 1). Notes: (1) be careful: invoking this routine will destroy the possibly information about your group (but will recompute it again) (2) users need not call this function normally for the solution of the word problem.
initializeGroupForWordProblem(gp, m, n) initializes the group gp for the word problem. Notes: (1) with a small integer you get shorter words, but the routine takes longer than the standard routine for longer words. (2) be careful: invoking this routine will destroy the possibly stored information about your group (but will recompute it again). (3) users need not call this function normally for the solution of the word problem.
member?(pp, gp) answers the question, whether the permutation pp is in the group gp or not.
movedPoints(gp) returns the points moved by the group gp.
orbit(gp, el) returns the orbit of the element el under the group gp, i.e. the set of all points gained by applying each group element to el.
orbit(gp, ls) returns the orbit of the ordered list ls under the group gp. Note: return type is L L S temporarily because FSET L S has an error.
orbit(gp, els) returns the orbit of the unordered set els under the group gp.
orbits(gp) returns the orbits of the group gp, i.e. it partitions the (finite) of all moved points.
order(gp) returns the order of the group gp.
perm_to_vec(supp, p, degree) should be local but conditional
permutationGroup(ls) coerces a list of permutations ls to the group generated by this list.
pointList(gp) should be local but conditional
random(gp) returns a random product of maximal 20 generators of the group gp. Note: random(gp)=random(gp, 20).
random(gp, i) returns a random product of maximal i generators of the group gp.
relationsInGenerators(gp) computes relations between generators.
relationsInStrongGenerators(gp) computes relations between strong generators.
strongGenerators(gp) returns strong generators for the group gp.
wordInGenerators(p, gp) returns the word for the permutation p in the original generators of the group gp, represented by the indices of the list, given by generators.
wordInStrongGenerators(p, gp) returns the word for the permutation p in the strong generators of the group gp, represented by the indices of the list, given by strongGenerators.
wordsForStrongGenerators(gp) returns the words for the strong generators of the group gp in the original generators of gp, represented by their indices in the list, given by generators.