# PermutationGroup SΒΆ

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 Language 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.

<=: (%, %) -> Boolean

`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.

<: (%, %) -> Boolean

`gp1 < gp2` returns `true` if and only if gp1 is a proper subgroup of gp2.

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

base: % -> List S

`base(gp)` returns a base for the group gp.

coerce: % -> List Permutation S

`coerce(gp)` returns the generators of the group gp.

coerce: % -> OutputForm
coerce: List Permutation S -> %

`coerce(ls)` coerces a list of permutations ls to the group generated by this list.

degree: % -> NonNegativeInteger

`degree(gp)` returns the number of points moved by all permutations of the group gp.

elt: (%, NonNegativeInteger) -> Permutation S

`elt(gp, i)` returns the `i`-th generator of the group gp.

generators: % -> List Permutation S

`generators(gp)` returns the generators of the group gp.

initializeGroupForWordProblem: % -> Void

`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 soultion of the word problem.

initializeGroupForWordProblem: (%, Integer, Integer) -> Void

`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 soultion of the word problem.

latex: % -> String

from SetCategory

member?: (Permutation S, %) -> Boolean

`member?(pp, gp)` answers the question, whether the permutation pp is in the group gp or not.

movedPoints: % -> Set S

`movedPoints(gp)` returns the points moved by the group gp.

orbit: (%, List S) -> Set List S

`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: (%, S) -> Set S

`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: (%, Set S) -> Set Set S

`orbit(gp, els)` returns the orbit of the unordered set els under the group gp.

orbits: % -> Set Set S

`orbits(gp)` returns the orbits of the group gp, i.e. it partitions the (finite) of all moved points.

order: % -> NonNegativeInteger

`order(gp)` returns the order of the group gp.

perm_to_vec: (List S, Permutation S, NonNegativeInteger) -> Vector NonNegativeInteger

`perm_to_vec(supp, p, degree)` should be local but conditional

permutationGroup: List Permutation S -> %

`permutationGroup(ls)` coerces a list of permutations ls to the group generated by this list.

pointList: % -> List S

`pointList(gp)` should be local but conditional

random: % -> Permutation S

`random(gp)` returns a random product of maximal 20 generators of the group gp. Note: random(gp)=random(gp, 20).

random: (%, Integer) -> Permutation S

`random(gp, i)` returns a random product of maximal `i` generators of the group gp.

relationsInGenerators: % -> List List Integer

`relationsInGenerators(gp)` computes relations between generators.

relationsInStrongGenerators: % -> List List Integer

`relationsInStrongGenerators(gp)` computes relations between strong generators.

strongGenerators: % -> List Permutation S

`strongGenerators(gp)` returns strong generators for the group gp.

wordInGenerators: (Permutation S, %) -> List NonNegativeInteger

`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: (Permutation S, %) -> List NonNegativeInteger

`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: % -> List List NonNegativeInteger

`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.

BasicType

SetCategory