Group homomorphisms

In OSCAR, a group homomorphism from G to H is an object of parametric type GAPGroupHomomorphism{S,T}, where S and T are the types of G and H respectively.

A homomorphism from G to H can be defined in two ways.

• Writing explicitly the images of the generators of G:
f = hom(G,H,[x1,x2,...],[y1,y2,...])

Here, [x1,x2,...] must be a generating set for G (not necessarily minimal) and [y1,y2,...] is a vector of elements of H of the same length of [x1,x2,...]. This assigns to f the value of the group homomorphism sending x_i into y_i.

An exception is thrown if such a homomorphism does not exist.

• Taking an existing function g satisfying the group homomorphism properties:
f = hom(G,H,g)

An exception is thrown if the function g does not define a group homomorphism.

Example: The following procedures define the same homomorphism (conjugation by x) in the two ways explained above.

julia> S=symmetric_group(4);

julia> x=S[1];

julia> f=hom(S,S,gens(S),[S[1]^x,S[2]^x]);

julia> g=hom(S,S,y->y^x);

julia> f==g
true
homMethod
hom(G::GAPGroup, H::GAPGroup, f::Function)

Return the group homomorphism defined by the function f.

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homMethod
hom(G::GAPGroup, H::GAPGroup, gensG::Vector = gens(G), imgs::Vector; check::Bool = true)

Return the group homomorphism defined by gensG[i] -> imgs[i] for every i. In order to work, the elements of gensG must generate G.

If check is set to false then it is not checked whether the mapping defines a group homomorphism.

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imageMethod
image(f::GAPGroupHomomorphism, x::GAPGroupElem)
(f::GAPGroupHomomorphism)(x::GAPGroupElem)

Return f(x).

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restrict_homomorphismMethod
restrict_homomorphism(f::GAPGroupHomomorphism, H::Group)
restrict_homomorphism(f::GAPGroupElem{AutomorphismGroup{T}}, H::T) where T <: Group

Return the restriction of f to H. An exception is thrown if H is not a subgroup of domain(f).

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OSCAR has also the following standard homomorphism.

trivial_morphismFunction
trivial_morphism(T::TorQuadModule, U::TorQuadModule) -> TorQuadModuleMap

Return the abelian group homomorphism between T and U sending every elements of T to the zero element of U.

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trivial_morphism(T::TorQuadModule) -> TorQuadModuleMap

Return the abelian group endomorphism of T sending every elements of T to the zero element of T.

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trivial_morphism(G::GAPGroup, H::GAPGroup = G)

Return the homomorphism from G to H sending every element of G into the identity of H.

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To evaluate the homomorphism f in the element x of G, it is possible to use the instruction

image(f,x)

or the more compact notations f(x) and x^f.

Example:

julia> S=symmetric_group(4);

julia> f=hom(S,S,x->x^S[1]);

julia> x=cperm(S,[1,2]);

julia> image(f,x)
(2,3)

julia> f(x)
(2,3)

julia> x^f
(2,3)

A sort of "inverse" of the evaluation is the following

has_preimage_with_preimageMethod
has_preimage_with_preimage(f::GAPGroupHomomorphism, x::GAPGroupElem; check::Bool = true)

Return (true, y) if there exists y in domain(f) such that f(y) = x holds; otherwise, return (false, o) where o is the identity of domain(f).

If check is set to false then the test whether x is an element of image(f) is omitted.

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Example:

julia> S=symmetric_group(4);

julia> f=hom(S,S,x->x^S[1]);

julia> x=cperm(S,[1,2]);

julia> has_preimage_with_preimage(f,x)
(true, (1,4))

Operations on homomorphisms

OSCAR supports the following operations on homomorphisms.

• inv(f) = the inverse of f. An exception is thrown if f is not bijective.

• f^n = the homomorphism f composed n times with itself. An exception is thrown if the domain and the codomain of f do not coincide (unless n=1). If n is negative, the result is the inverse of f composed n times with itself.

• compose(f, g) = composition of f and g. This works only if the codomain of f coincides with the domain of g. Shorter equivalent expressions are f*g and g(f).

Example:

julia> S=symmetric_group(4);

julia> f=hom(S,S,x->x^S[1]);

julia> g=hom(S,S,x->x^S[2]);

julia> f*g==hom(S,S,x->x^(S[1]*S[2]))
true

julia> f==f^-3
true
Note

The composition operation * has to be read from the right to the left. So, (f*g)(x) is equivalent to g(f(x)).

Properties of homomorphisms

OSCAR implements the following attributes of homomorphisms, in addition to the usual domain and codomain.

is_invariantMethod
is_invariant(f::GAPGroupHomomorphism, H::Group)
is_invariant(f::GAPGroupElem{AutomorphismGroup{T}}, H::T)

Return whether f(H) == H holds. An exception is thrown if domain(f) and codomain(f) are not equal or if H is not contained in domain(f).

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Subgroups described by homomorphisms

The following functions compute subgroups or quotients of either the domain or the codomain. Analogously to the functions described in Sections Subgroups and Quotients, the output consists of a pair (H, g), where H is a subgroup (resp. quotient) and g is its embedding (resp. projection) homomorphism.

kernelMethod
kernel(f::GAPGroupHomomorphism)

Return the kernel of f, together with its embedding into domain(f).

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imageMethod
image(f::GAPGroupHomomorphism)

Return the image of f as subgroup of codomain(f), together with the embedding homomorphism.

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imageMethod
image(f::GAPGroupHomomorphism{S, T}, H::S) where S <: GAPGroup where T <: GAPGroup
(f::GAPGroupHomomorphism{S, T})(H::S)

Return f(H), together with the embedding homomorphism into codomain(f).

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cokernelMethod
cokernel(f::GAPGroupHomomorphism)

Return the cokernel of f, that is, the quotient of the codomain of f by the normal closure of the image.

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preimageMethod
preimage(f::GAPGroupHomomorphism{S, T}, H::GAPGroup) where S <: GAPGroup where T <: GAPGroup

If H is a subgroup of the codomain of f, return the subgroup f^-1(H), together with its embedding homomorphism into the domain of f.

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Group isomorphisms

For all functions that return group isomorphisms, we have the following rule about the direction of the result.

If two groups are given as inputs then the domain of the returned isomorphism is the first given group and the codomain is the second.

If one group is given then the domain of the result is this group, and the codomain is some new group constructed by the function.

is_isomorphicMethod
is_isomorphic(G::Group, H::Group)

Return true if G and H are isomorphic groups, and false otherwise.

Examples

julia> is_isomorphic(symmetric_group(3), dihedral_group(6))
true
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is_isomorphic_with_mapMethod
is_isomorphic_with_map(G::Group, H::Group)

Return (true,f) if G and H are isomorphic groups, where f is a group isomorphism. Otherwise, return (false,f), where f is the trivial homomorphism.

Examples

julia> is_isomorphic_with_map(symmetric_group(3), dihedral_group(6))
(true, Hom: Sym(3) -> pc group)
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isomorphismMethod
isomorphism(G::Group, H::Group)

Return a group isomorphism between G and H if they are isomorphic groups. Otherwise throw an exception.

Examples

julia> isomorphism(symmetric_group(3), dihedral_group(6))
Group homomorphism
from Sym(3)
to pc group of order 6
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isomorphismMethod
isomorphism(::Type{T}, G::GAPGroup) where T <: Union{SubPcGroup, SubFPGroup, PermGroup}
isomorphism(::Type{T}, G::GAPGroup; on_gens=false) where T <: Union{PcGroup, FPGroup}

Return an isomorphism from G to a group H of type T. An exception is thrown if no such isomorphism exists.

If on_gens is true then gens(G) is guaranteed to correspond to gens(H); an exception is thrown if this is not possible.

Isomorphisms are cached in G, subsequent calls of isomorphism with the same T (and the same value of on_gens) yield identical results.

If only the image of such an isomorphism is needed, use T(G).

Examples

julia> G = dihedral_group(6)
Pc group of order 6

julia> iso = isomorphism(PermGroup, G)
Group homomorphism
from pc group of order 6
to permutation group of degree 6 and order 6

julia> permutation_group(G)
Permutation group of degree 6 and order 6

julia> codomain(iso) === ans
true
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isomorphismMethod
isomorphism(::Type{FinGenAbGroup}, G::GAPGroup)

Return a map from G to an isomorphic (additive) group of type FinGenAbGroup. An exception is thrown if G is not abelian or not finite.

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simplified_fp_groupMethod
simplified_fp_group(G::FPGroup)

Return a group H of type FPGroup and an isomorphism f from G to H, where the presentation of H was obtained from the presentation of G by applying Tietze transformations in order to reduce it with respect to the number of generators, the number of relators, and the relator lengths.

Examples

julia> F = free_group(3)
Free group of rank 3

julia> G = quo(F, [gen(F,1)])[1]
Finitely presented group of infinite order

julia> simplified_fp_group(G)[1]
Finitely presented group of infinite order
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Other homomorphisms

epimorphism_from_free_groupMethod
epimorphism_from_free_group(G::GAPGroup)

Return an epimorphism epi from a free group F == domain(epi) onto G, where F has the same number of generators as G and such that for each i it maps gen(F,i) to gen(G,i).

A useful application of this function is expressing an element of G as a word in its generators.

Examples

julia> G = symmetric_group(4);

julia> epi = epimorphism_from_free_group(G)
Group homomorphism
from free group of rank 2
to Sym(4)

julia> pi = G([2,4,3,1])
(1,2,4)

julia> w = preimage(epi, pi);

julia> map_word(w, gens(G))
(1,2,4)
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