Group homomorphisms for groups with a GAP backend¶
EXAMPLES:
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: A = AbelianGroupGap([2, 4])
sage: F.<a,b> = FreeGroup()
sage: f = F.hom([g for g in A.gens()])
sage: K = f.kernel()
sage: K
Group(<free, no generators known>)
AUTHORS:
 Simon Brandhorst (20180208): initial version
 Sebastian Oehms (20181115): have this functionality work for permutation groups (trac ticket #26750)
and implement
section()
andnatural_map()

class
sage.groups.libgap_morphism.
GroupHomset_libgap
(G, H, category=None, check=True)¶ Bases:
sage.categories.homset.HomsetWithBase
Homsets of groups with a libgap backend.
Do not call this directly instead use
Hom()
.INPUT:
G
– a libgap groupH
– a libgap groupcategory
– a category
OUTPUT:
The homset of two libgap groups.
EXAMPLES:
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap sage: A = AbelianGroupGap([2,4]) sage: H = A.Hom(A) sage: H Set of Morphisms from Abelian group with gap, generator orders (2, 4) to Abelian group with gap, generator orders (2, 4) in Category of finite enumerated commutative groups

Element
¶ alias of
GroupMorphism_libgap

natural_map
()¶ This method from
HomsetWithBase
is overloaded here for cases in which both groups have corresponding lists of generators.OUTPUT:
an instance of the element class of self if there exists a group homomorphism mapping the generators of the domain of self to the according generators of the codomain. Else the method falls back to the default.
EXAMPLES:
sage: G = GL(3,2) sage: P = PGL(3,2) sage: nat = Hom(G, P).natural_map() sage: type(nat) <class 'sage.groups.libgap_morphism.GroupHomset_libgap_with_category.element_class'> sage: g1, g2 = G.gens() sage: nat(g1*g2) (1,2,4,5,7,3,6)

class
sage.groups.libgap_morphism.
GroupMorphism_libgap
(homset, gap_hom, check=True)¶ Bases:
sage.categories.morphism.Morphism
This wraps GAP group homomorphisms.
Checking if the input defines a group homomorphism can be expensive if the group is large.
INPUT:
homset
– the parentgap_hom
– asage.libs.gap.element.GapElement
consisting of a group homomorphismcheck
– (default:True
) check if thegap_hom
is a group homomorphism; this can be expensive
EXAMPLES:
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap sage: A = AbelianGroupGap([2, 4]) sage: A.hom([g^2 for g in A.gens()]) Group endomorphism of Abelian group with gap, generator orders (2, 4)
Homomorphisms can be defined between different kinds of GAP groups:
sage: G = MatrixGroup([Matrix(ZZ, 2, [0,1,1,0])]) sage: f = A.hom([G.0, G(1)]) sage: f Group morphism: From: Abelian group with gap, generator orders (2, 4) To: Matrix group over Integer Ring with 1 generators ( [0 1] [1 0] ) sage: G.<a,b> = FreeGroup() sage: H = G / (G([1]), G([2])^3) sage: f = G.hom(H.gens()) sage: f Group morphism: From: Free Group on generators {a, b} To: Finitely presented group < a, b  a, b^3 >
Homomorphisms can be defined between GAP groups and permutation groups:
sage: S = Sp(4,3) sage: P = PSp(4,3) sage: pr = S.hom(P.gens()) sage: E = copy(S.one().matrix()) sage: E[3,0] = 2; e = S(E) sage: pr(e) (1,16,15)(3,22,18)(4,19,21)(6,34,24)(7,25,33)(9,40,27)(10,28,39)(12,37,30)(13,31,36)

gap
()¶ Return the underlying LibGAP group homomorphism.
EXAMPLES:
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap sage: A = AbelianGroupGap([2,4]) sage: f = A.hom([g^2 for g in A.gens()]) sage: f.gap() [ f1, f2 ] > [ <identity> of ..., f3 ]

image
(J, *args, **kwds)¶ The image of an element or a subgroup.
INPUT:
J
– a subgroup or an element of the domain ofself
OUTPUT:
The image of
J
underself
.Note
pushforward
is the method that is used when a map is called on anything that is not an element of its domain. For historical reasons, we keep the aliasimage()
for this method.EXAMPLES:
sage: G.<a,b> = FreeGroup() sage: H = G / (G([1]), G([2])^3) sage: f = G.hom(H.gens()) sage: S = G.subgroup([a.gap()]) sage: f.pushforward(S) Group([ a ]) sage: x = f.image(a) sage: x a sage: x.parent() Finitely presented group < a, b  a, b^3 > sage: G = GU(3,2) sage: P = PGU(3,2) sage: pr = Hom(G, P).natural_map() sage: GS = G.subgroup([G.gen(0)]) sage: pr.pushforward(GS) Subgroup generated by [(3,4,5)(10,18,14)(11,19,15)(12,20,16)(13,21,17)] of (The projective general unitary group of degree 3 over Finite Field of size 2)

kernel
()¶ Return the kernel of
self
.EXAMPLES:
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap sage: A1 = AbelianGroupGap([6, 6]) sage: A2 = AbelianGroupGap([3, 3]) sage: f = A1.hom(A2.gens()) sage: f.kernel() Subgroup of Abelian group with gap, generator orders (6, 6) generated by (f1*f2, f3*f4) sage: f.kernel().order() 4 sage: S = Sp(6,3) sage: P = PSp(6,3) sage: pr = Hom(S, P).natural_map() sage: pr.kernel() Subgroup with 1 generators ( [2 0 0 0 0 0] [0 2 0 0 0 0] [0 0 2 0 0 0] [0 0 0 2 0 0] [0 0 0 0 2 0] [0 0 0 0 0 2] ) of Symplectic Group of degree 6 over Finite Field of size 3

lift
(h)¶ Return an element of the domain that maps to
h
.EXAMPLES:
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap sage: A = AbelianGroupGap([2,4]) sage: f = A.hom([g^2 for g in A.gens()]) sage: a = A.gens()[1] sage: f.lift(a^2) f2
If the element is not in the image, we raise an error:
sage: f.lift(a) Traceback (most recent call last): ... ValueError: f2 is not an element of the image of Group endomorphism of Abelian group with gap, generator orders (2, 4)

preimage
(S)¶ Return the preimage of the subgroup
S
.INPUT:
S
– a subgroup of this group
EXAMPLES:
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap sage: A = AbelianGroupGap([2,4]) sage: B = AbelianGroupGap([4]) sage: f = A.hom([B.one(), B.gen(0)^2]) sage: S = B.subgroup([B.one()]) sage: f.preimage(S) == f.kernel() True sage: S = Sp(4,3) sage: P = PSp(4,3) sage: pr = Hom(S, P).natural_map() sage: PS = P.subgroup([P.gen(0)]) sage: pr.preimage(PS) Subgroup with 2 generators ( [2 0 0 0] [1 0 0 0] [0 2 0 0] [0 2 0 0] [0 0 2 0] [0 0 2 0] [0 0 0 2], [0 0 0 1] ) of Symplectic Group of degree 4 over Finite Field of size 3

pushforward
(J, *args, **kwds)¶ The image of an element or a subgroup.
INPUT:
J
– a subgroup or an element of the domain ofself
OUTPUT:
The image of
J
underself
.Note
pushforward
is the method that is used when a map is called on anything that is not an element of its domain. For historical reasons, we keep the aliasimage()
for this method.EXAMPLES:
sage: G.<a,b> = FreeGroup() sage: H = G / (G([1]), G([2])^3) sage: f = G.hom(H.gens()) sage: S = G.subgroup([a.gap()]) sage: f.pushforward(S) Group([ a ]) sage: x = f.image(a) sage: x a sage: x.parent() Finitely presented group < a, b  a, b^3 > sage: G = GU(3,2) sage: P = PGU(3,2) sage: pr = Hom(G, P).natural_map() sage: GS = G.subgroup([G.gen(0)]) sage: pr.pushforward(GS) Subgroup generated by [(3,4,5)(10,18,14)(11,19,15)(12,20,16)(13,21,17)] of (The projective general unitary group of degree 3 over Finite Field of size 2)

section
()¶ This method returns a section map of self by use of
lift()
. Seesection()
ofsage.categories.map.Map
, as well.OUTPUT:
an instance of
sage.categories.morphism.SetMorphism
mapping an element of the codomain of self to one of its preimagesEXAMPLES:
sage: G = GU(3,2) sage: P = PGU(3,2) sage: pr = Hom(G, P).natural_map() sage: sect = pr.section() sage: sect(P.an_element()) [a + 1 a a] [ 1 1 0] [ a 0 0]