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CurrentModule = Oscar
DocTestSetup = Oscar.doctestsetup()

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

hom(G::GAPGroup, H::GAPGroup, img::Function)
hom(G::GAPGroup, H::GAPGroup, gensG::Vector, imgs::Vector)
image(f::GAPGroupHomomorphism, x::GAPGroupElem)
preimage(f::GAPGroupHomomorphism, x::GAPGroupElem)
restrict_homomorphism(f::GAPGroupHomomorphism, H::GAPGroup)

OSCAR has also the following standard homomorphism.

id_hom
trivial_morphism

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

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_injective(f::GAPGroupHomomorphism)
is_surjective(f::GAPGroupHomomorphism)
is_bijective(f::GAPGroupHomomorphism)
is_invertible(f::GAPGroupHomomorphism)
is_invariant(f::GAPGroupHomomorphism, H::GAPGroup)

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](@ref subgroups) and [Quotients](@ref quotient), the output consists of a pair (H, g), where H is a subgroup (resp. quotient) and g is its embedding (resp. projection) homomorphism.

kernel(f::GAPGroupHomomorphism)
image(f::GAPGroupHomomorphism)
image(f::GAPGroupHomomorphism{S, T}, H::S) where S <: GAPGroup where T <: GAPGroup
cokernel(f::GAPGroupHomomorphism)
preimage(f::GAPGroupHomomorphism{S, T}, H::T) where S <: GAPGroup where T <: GAPGroup

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_isomorphic(G::GAPGroup, H::GAPGroup)
is_isomorphic_with_map(G::GAPGroup, H::GAPGroup)
isomorphism(G::GAPGroup, H::GAPGroup)

isomorphism(::Type{T}, G::GAPGroup) where T <: Union{SubPcGroup, PermGroup}
isomorphism(::Type{FinGenAbGroup}, G::GAPGroup)
simplified_fp_group(G::FPGroup)

Other homomorphisms

epimorphism_from_free_group(G::GAPGroup)