Coverage for local/lib/python2.7/site-packages/sage/groups/perm_gps/permgroup_morphism.py : 51%

r""" 

Permutation group homomorphisms 

AUTHORS: 

- David Joyner (2006-03-21): first version 

- David Joyner (2008-06): fixed kernel and image to return a group, 

instead of a string. 

EXAMPLES:: 

sage: G = CyclicPermutationGroup(4) 

sage: H = DihedralGroup(4) 

sage: g = G([(1,2,3,4)]) 

sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())) 

sage: phi.image(G) 

Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)] 

sage: phi.kernel() 

Subgroup of (Cyclic group of order 4 as a permutation group) generated by [()] 

sage: phi.image(g) 

(1,2,3,4) 

sage: phi(g) 

(1,2,3,4) 

sage: phi.codomain() 

Dihedral group of order 8 as a permutation group 

sage: phi.codomain() 

Dihedral group of order 8 as a permutation group 

sage: phi.domain() 

Cyclic group of order 4 as a permutation group 

""" 

#***************************************************************************** 

# Copyright (C) 2006 David Joyner and William Stein <wstein@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.categories.morphism import Morphism 

from sage.groups.perm_gps.permgroup import PermutationGroup, PermutationGroup_generic 

class PermutationGroupMorphism(Morphism): 

""" 

A set-theoretic map between PermutationGroups. 

""" 

def _repr_type(self): 

""" 

Returns the type of this morphism. This is used for printing 

the morphism. 

EXAMPLES:: 

sage: G = PSL(2,7) 

sage: D, iota1, iota2, pr1, pr2 = G.direct_product(G) 

sage: pr1._repr_type() 

'Permutation group' 

""" 

return "Permutation group" 

def kernel(self): 

""" 

Returns the kernel of this homomorphism as a permutation group. 

EXAMPLES:: 

sage: G = CyclicPermutationGroup(4) 

sage: H = DihedralGroup(4) 

sage: g = G([(1,2,3,4)]) 

sage: phi = PermutationGroupMorphism_im_gens(G, H, [1]) 

sage: phi.kernel() 

Subgroup of (Cyclic group of order 4 as a permutation group) generated by [(1,2,3,4)] 

:: 

sage: G = PSL(2,7) 

sage: D = G.direct_product(G) 

sage: H = D[0] 

sage: pr1 = D[3] 

sage: G.is_isomorphic(pr1.kernel()) 

True 

""" 

return self.domain().subgroup(gap_group=self._gap_().Kernel()) 

def image(self, J): 

""" 

J must be a subgroup of G. Computes the subgroup of H which is the 

image of J. 

EXAMPLES:: 

sage: G = CyclicPermutationGroup(4) 

sage: H = DihedralGroup(4) 

sage: g = G([(1,2,3,4)]) 

sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())) 

sage: phi.image(G) 

Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)] 

sage: phi.image(g) 

(1,2,3,4) 

:: 

sage: G = PSL(2,7) 

sage: D = G.direct_product(G) 

sage: H = D[0] 

sage: pr1 = D[3] 

sage: pr1.image(G) 

Subgroup of (The projective special linear group of degree 2 over Finite Field of size 7) generated by [(3,7,5)(4,8,6), (1,2,6)(3,4,8)] 

sage: G.is_isomorphic(pr1.image(G)) 

True 

""" 

H = self.codomain() 

if J in self.domain(): 

J = PermutationGroup([J]) 

G = self._gap_().Image(J) 

return H.subgroup(gap_group=G).gens()[0] 

else: 

G = self._gap_().Image(J) 

return H.subgroup(gap_group=G) 

def __call__(self, g): 

""" 

Some python code for wrapping GAP's Images function but only for 

permutation groups. Returns an error if g is not in G. 

EXAMPLES:: 

sage: G = CyclicPermutationGroup(4) 

sage: H = DihedralGroup(4) 

sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())) 

sage: g = G([(1,3),(2,4)]); g 

(1,3)(2,4) 

sage: phi(g) 

(1,3)(2,4) 

""" 

return self.image(g) 

class PermutationGroupMorphism_id(PermutationGroupMorphism): 

pass 

class PermutationGroupMorphism_from_gap(PermutationGroupMorphism): 

def __init__(self, G, H, gap_hom): 

""" 

This is a Python trick to allow Sage programmers to create a group 

homomorphism using GAP using very general constructions. An example 

of its usage is in the direct_product instance method of the 

PermutationGroup_generic class in permgroup.py. 

Basic syntax: 

PermutationGroupMorphism_from_gap(domain_group, 

range_group,'phi:=gap_hom_command;','phi') And don't forget the 

line: from sage.groups.perm_gps.permgroup_morphism import 

PermutationGroupMorphism_from_gap in your program. 

EXAMPLES:: 

sage: from sage.groups.perm_gps.permgroup_morphism import PermutationGroupMorphism_from_gap 

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) 

sage: H = G.subgroup([G([(1,2,3,4)])]) 

sage: PermutationGroupMorphism_from_gap(H, G, gap.Identity) 

Permutation group morphism: 

From: Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2,3,4)] 

To: Permutation Group with generators [(1,2)(3,4), (1,2,3,4)] 

Defn: Identity 

""" 

if not all(isinstance(X, PermutationGroup_generic) for X in [G, H]): 

raise TypeError("Sorry, the groups must be permutation groups.") 

PermutationGroupMorphism.__init__(self, G, H) 

self._gap_hom = gap_hom 

def _repr_defn(self): 

""" 

Returns the definition of this morphism. This is used when 

printing the morphism. 

EXAMPLES:: 

sage: from sage.groups.perm_gps.permgroup_morphism import PermutationGroupMorphism_from_gap 

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) 

sage: H = G.subgroup([G([(1,2,3,4)])]) 

sage: phi = PermutationGroupMorphism_from_gap(H, G, gap.Identity) 

sage: phi._repr_defn() 

'Identity' 

""" 

return str(self._gap_hom).replace('\n', '') 

def _gap_(self, gap=None): 

""" 

Returns a GAP version of this morphism. 

EXAMPLES:: 

sage: from sage.groups.perm_gps.permgroup_morphism import PermutationGroupMorphism_from_gap 

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) 

sage: H = G.subgroup([G([(1,2,3,4)])]) 

sage: phi = PermutationGroupMorphism_from_gap(H, G, gap.Identity) 

sage: phi._gap_() 

Identity 

""" 

return self._gap_hom 

def __call__(self, g): 

""" 

Some python code for wrapping GAP's Images function but only for 

permutation groups. Returns an error if g is not in G. 

EXAMPLES:: 

sage: G = PSL(2,7) 

sage: D = G.direct_product(G) 

sage: H = D[0] 

sage: pr1 = D[3] 

sage: [pr1(g) for g in G.gens()] 

[(3,7,5)(4,8,6), (1,2,6)(3,4,8)] 

""" 

return self.codomain()(self._gap_().Image(g)) 

class PermutationGroupMorphism_im_gens(PermutationGroupMorphism): 

def __init__(self, G, H, gens=None): 

""" 

Some python code for wrapping GAP's GroupHomomorphismByImages 

function but only for permutation groups. Can be expensive if G is 

large. Returns "fail" if gens does not generate self or if the map 

does not extend to a group homomorphism, self - other. 

EXAMPLES:: 

sage: G = CyclicPermutationGroup(4) 

sage: H = DihedralGroup(4) 

sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())); phi 

Permutation group morphism: 

From: Cyclic group of order 4 as a permutation group 

To: Dihedral group of order 8 as a permutation group 

Defn: [(1,2,3,4)] -> [(1,2,3,4)] 

sage: g = G([(1,3),(2,4)]); g 

(1,3)(2,4) 

sage: phi(g) 

(1,3)(2,4) 

sage: images = ((4,3,2,1),) 

sage: phi = PermutationGroupMorphism_im_gens(G, G, images) 

sage: g = G([(1,2,3,4)]); g 

(1,2,3,4) 

sage: phi(g) 

(1,4,3,2) 

AUTHORS: 

- David Joyner (2006-02) 

""" 

if not all([isinstance(X, PermutationGroup_generic) for X in [G, H]]): 

raise TypeError("Sorry, the groups must be permutation groups.") 

PermutationGroupMorphism.__init__(self, G, H) 

self._images = [H(img) for img in gens] 

def _repr_defn(self): 

""" 

Returns the definition of this morphism. This is used when 

printing the morphism. 

EXAMPLES:: 

sage: G = CyclicPermutationGroup(4) 

sage: H = DihedralGroup(4) 

sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())) 

sage: phi._repr_defn() 

'[(1,2,3,4)] -> [(1,2,3,4)]' 

""" 

return "%s -> %s"%(self.domain().gens(), self._images) 

def _gap_(self): 

""" 

Returns a GAP representation of this morphism. 

EXAMPLES:: 

sage: G = CyclicPermutationGroup(4) 

sage: H = DihedralGroup(4) 

sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())) 

sage: phi._gap_() 

GroupHomomorphismByImages( Group( [ (1,2,3,4) ] ), Group( 

[ (1,2,3,4), (1,4)(2,3) ] ), [ (1,2,3,4) ], [ (1,2,3,4) ] ) 

""" 

return self.domain()._gap_().GroupHomomorphismByImages(self.codomain(), self.domain().gens(), self._images) 

def is_PermutationGroupMorphism(f): 

""" 

Returns True if the argument ``f`` is a PermutationGroupMorphism. 

EXAMPLES:: 

sage: from sage.groups.perm_gps.permgroup_morphism import is_PermutationGroupMorphism 

sage: G = CyclicPermutationGroup(4) 

sage: H = DihedralGroup(4) 

sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())) 

sage: is_PermutationGroupMorphism(phi) 

True 

""" 

return isinstance(f, PermutationGroupMorphism)