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)
Symmetric group of degree 4

julia> x = S[1]
(1,2,3,4)

julia> f = hom(S, S, [S[1]^x, S[2]^x])
Group homomorphism
  from symmetric group of degree 4
  to symmetric group of degree 4

julia> g = hom(S, S, y -> y^x)
Group homomorphism
  from symmetric group of degree 4
  to symmetric group of degree 4

julia> f == g
true

hom — Method

hom(G::GAPGroup, H::GAPGroup, f::Function)

Return the group homomorphism defined by the function f.

hom — Method

hom(G::Group, H::Group, 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.

image — Method

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

Return f(x).

preimage — Method

preimage(f::GAPGroupHomomorphism, x::GAPGroupElem)

Return an element y in the domain of f with the property f(y) == x. See has_preimage_with_preimage(f::GAPGroupHomomorphism, x::GAPGroupElem; check::Bool = true) for a check whether x has such a preimage.

restrict_homomorphism — Method

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).

OSCAR has also the following standard homomorphism.

id_hom — Method

id_hom(G::GAPGroup)

Return the identity homomorphism on the group G.

trivial_morphism — Function

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

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

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)
Symmetric group of degree 4

julia> f = hom(S, S, x->x^S[1])
Group homomorphism
  from symmetric group of degree 4
  to symmetric group of degree 4

julia> x = cperm(S, [1,2])
(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 — Method

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.

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)
Symmetric group of degree 4

julia> f = hom(S, S, x->x^S[1])
Group homomorphism
  from symmetric group of degree 4
  to symmetric group of degree 4

julia> g = hom(S, S, x->x^S[2])
Group homomorphism
  from symmetric group of degree 4
  to symmetric group of degree 4

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 — Method

is_injective(f::GAPGroupHomomorphism)

Return whether f is injective.

is_surjective — Method

is_surjective(f::GAPGroupHomomorphism)

Return whether f is surjective.

is_bijective — Method

is_bijective(f::GAPGroupHomomorphism)

Return whether f is bijective.

is_invertible — Method

is_invertible(f::GAPGroupHomomorphism)

Return whether f is invertible.

is_invariant — Method

is_invariant(f::GAPGroupHomomorphism, H::GAPGroup)

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).

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.

kernel — Method

kernel(f::GAPGroupHomomorphism)

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

image — Method

image(f::GAPGroupHomomorphism)

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

image — Method

image(f::GAPGroupHomomorphism{<:GAPGroup, <:GAPGroup}, H::GAPGroup)
(f::GAPGroupHomomorphism{<:GAPGroup, <:GAPGroup})(H::GAPGroup)

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

cokernel — Method

cokernel(f::GAPGroupHomomorphism)

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

preimage — Method

preimage(f::GAPGroupHomomorphism{<: GAPGroup, <: GAPGroup}, H::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.

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 — Method

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

is_isomorphic_with_map — Method

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)

isomorphism — Method

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 symmetric group of degree 3
  to pc group of order 6

isomorphic_subgroups — Method

isomorphic_subgroups(H::Group, G::Group)

Return a vector of injective group homomorphism from H to G, where the images are representatives of those conjugacy classes of subgroups of G that are isomorphic with H.

Examples

julia> isomorphic_subgroups(alternating_group(5), alternating_group(6))
2-element Vector{GAPGroupHomomorphism{PermGroup, PermGroup}}:
 Hom: Alt(5) -> Alt(6)
 Hom: Alt(5) -> Alt(6)

julia> isomorphic_subgroups(symmetric_group(4), alternating_group(5))
GAPGroupHomomorphism{PermGroup, PermGroup}[]

isomorphism — Method

isomorphism(::Type{T}, G::Group; on_gens=false) where T <: Group

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 usually 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); but this will assume on_gens=false.

isomorphism — Method

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.