AssociationSchemes.permutation_group module

class AssociationSchemes.permutation_group.PermutationGroup(gens=None, gap_group=None, canonicalize=True, domain=None, category=None)

Bases: PermutationGroup_generic

action_on_subsets(k)

Return the action of self on the k-sets of self.domain().

EXAMPLE:

sage: G = PermutationGroup(SymmetricGroup(5).gens())
sage: K = G.action_on_subsets(2)
sage: K.is_transitive()
True

group_action(H)

Return the action of self on the cosets of the subgroup H of self by right multiplication.

EXAMPLE:

sage: U = PSL(2,7)
sage: G = PermutationGroup(U.gens())
sage: M = G.conjugacy_classes_subgroups()
sage: H = M[5]
sage: H.structure_description()
'C4'
sage: K = G.group_action(H)
sage: K.structure_description()
'PSL(3,2)'
sage: K.stabilizer(K.domain()[0]).structure_description()
'C4'

is_core_free(H)

is_derangement(x)

Return whether self is a derangement.

EXAMPLE:

sage: G = PermutationGroup(SymmetricGroup(5).gens())
sage: G.is_derangement(G((1,5,2)))
False
sage: G.is_derangement(G("(1,5,2)(3,4)"))
True

is_quasi_primitive()

number_of_derangements()

Return the number of derangements in self.

orbital_digraphs()

permutation_character()

pointwise_stabilizer(S)

rank_of_group()

Return the rank of the permutation group self in its action on self.domain().

EXAMPLE:

sage: G = PermutationGroup(SymmetricGroup(5).gens())
sage: G.rank_of_group()
2
sage: K = G.action_on_subsets(2)
sage: K.rank_of_group()
3

setwise_stabilizer(S)

stabilizer_of_invariant_partition(L)

sub_orbits(v)