snf(A::GrpAbFinGen) -> GrpAbFinGen, GrpAbFinGenMap

Return a pair $(G, f)$, where $G$ is an abelian group in canonical Smith normal form isomorphic to $A$ and an isomorphism $f : G \to A$.

find_isomorphism(G, op, A::GrpAb) -> Dict, Dict

Given an abelian group $A$ and a collection $G$ which is an abelian group with the operation op, this functions returns isomorphisms $G \to A$ and $A \to G$ encoded as dictionaries.

It is assumed that $G$ and $A$ are isomorphic.

torsion_subgroup(G::GrpAbFinGen) -> GrpAbFinGen, Map

Return the torsion subgroup of G.

sub(G::GrpAbFinGen, s::Vector{GrpAbFinGenElem}) -> GrpAbFinGen, GrpAbFinGenMap

Create the subgroup $H$ of $G$ generated by the elements in s together with the injection $\iota : H \to G$.

sub(F::FreeMod{T}, V::Vector{<:FreeModElem{T}}, task::Symbol = :with_morphism) where T

Given a vector V of (homogeneous) elements of F, return the (graded) submodule of F generated by these elements.

Put more precisely, return the submodule as an object of type SubquoModule.

Additionally, if N denotes this object,

• return the inclusion map N $\to$ F if task = :with_morphism (default),
• return and cache the inclusion map N $\to$ F if task = :cache_morphism,
• do none of the above if task = :none.

If task = :only_morphism, return only the inclusion map.

sub(F::FreeMod{T}, A::MatElem{T}, task::Symbol = :with_morphism) where {T}

Given a (homogeneous) matrix A, return the (graded) submodule of F generated by the rows of A.

Put more precisely, return this submodule as an object of type SubquoModule.

Additionally, if N denotes this submodule,

• return the inclusion map N $\to$ F if task = :with_morphism (default),
• return and cache the inclusion map N $\to$ F if task = :cache_morphism,
• do none of the above if task = :none.

If task = :only_morphism, return only the inclusion map.

sub(F::FreeMod{T}, O::Vector{<:SubquoModuleElem{T}}, task::Symbol = :with_morphism) where T

Return S as a submodule of F, where S is generated by O. The embedding module of the parent of the elements of O must be F. If task is set to :none or to :module return only S. If task is set to :with_morphism (default option) or to :both return also the canonical injection morphism $S \to F$. If task is set to :cache_morphism the morphism is also cached. If task is set to :only_morphism return only the morphism.

sub(F::FreeMod{T}, s::SubquoModule{T}, task::Symbol = :with_morphism) where T

Return s as a submodule of F, that is the embedding free module of s must be F and s has no relations. If task is set to :none or to :module return only s. If task is set to :with_morphism (default option) or to :both return also the canonical injection morphism $s \to F$. If task is set to :cache_morphism the morphism is also cached. If task is set to :only_morphism return only the morphism.

sub(M::SubquoModule{T}, V::Vector{<:SubquoModuleElem{T}}, task::Symbol = :with_morphism) where T

Given a vector V of (homogeneous) elements of M, return the (graded) submodule of M generated by these elements.

Put more precisely, return this submodule as an object of type SubquoModule.

Additionally, if N denotes this object,

• return the inclusion map N $\to$ M if task = :with_morphism (default),
• return and cache the inclusion map N $\to$ M if task = :cache_morphism,
• do none of the above if task = :none.

If task = :only_morphism, return only the inclusion map.

sub(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}, task::Symbol = :with_morphism) where T

Given a vector V of (homogeneous) elements of M, return the (graded) submodule of M generated by these elements.

Put more precisely, return this submodule as an object of type SubquoModule.

Additionally, if N denotes this object,

• return the inclusion map N $\to$ M if task = :with_morphism (default),
• return and cache the inclusion map N $\to$ M if task = :cache_morphism,
• do none of the above if task = :none.

If task = :only_morphism, return only the inclusion map.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);

julia> F = free_module(R, 1);

julia> V = [x^2*F[1]; y^3*F[1]; z^4*F[1]];

julia> N, incl = sub(F, V);

julia> N
Submodule with 3 generators
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1]
represented as subquotient with no relations.

julia> incl
Map with following data
Domain:
=======
Submodule with 3 generators
1 -> x^2*e[1]
2 -> y^3*e[1]
3 -> z^4*e[1]
represented as subquotient with no relations.
Codomain:
=========
Free module of rank 1 over Multivariate polynomial ring in 3 variables over QQ

sub(A::SMat, r::UnitRange, c::UnitRange) -> SMat

Return the submatrix of $A$, where the rows correspond to $r$ and the columns correspond to $c$.

sub(s::Vector{GrpAbFinGenElem}) -> GrpAbFinGen, GrpAbFinGenMap

Assuming that the non-empty array s contains elements of an abelian group $G$, this functions returns the subgroup $H$ of $G$ generated by the elements in s together with the injection $\iota : H \to G$.

sub(G::GrpAbFinGen, M::ZZMatrix) -> GrpAbFinGen, GrpAbFinGenMap

Create the subgroup $H$ of $G$ generated by the elements corresponding to the rows of $M$ together with the injection $\iota : H \to G$.

sub(G::GrpAbFinGen, n::ZZRingElem) -> GrpAbFinGen, GrpAbFinGenMap

Create the subgroup $n \cdot G$ of $G$ together with the injection $\iota : n\cdot G \to G$.

sub(G::GrpAbFinGen, n::Integer) -> GrpAbFinGen, Map

Create the subgroup $n \cdot G$ of $G$ together with the injection $\iota : n \cdot G \to G$.

psylow_subgroup(G::GrpAbFinGen, p::IntegerUnion) -> GrpAbFinGen, GrpAbFinGenMap

Return the $p$-Sylow subgroup of G.

has_quotient(G::GrpAbFinGen, invariant::Vector{Int}) -> Bool

Given an abelian group $G$, return true if it has a quotient with given elementary divisors and false otherwise.

has_complement(f::GrpAbFinGenMap) -> Bool, GrpAbFinGenMap
has_complement(U::GrpAbFinGen, G::GrpAbFinGen) -> Bool, GrpAbFinGenMap

Given a map representing a subgroup of a group $G$, or a subgroup U of a group G, return either true and an injection of a complement in $G$, or false.

See also: is_pure

is_pure(U::GrpAbFinGen, G::GrpAbFinGen) -> Bool

A subgroup U of G is called pure if for all n an element in U that is in the image of the multiplication by n map of G is also a multiple of an element in U.

For finite abelian groups this is equivalent to U having a complement in G. They are also know as isolated subgroups and serving subgroups.

See also: is_neat, has_complement

EXAMPLES

julia> G = abelian_group([2, 8]);

julia> U, _ = sub(G, [G[1]+2*G[2]]);

julia> is_pure(U, G)
false

julia> U, _ = sub(G, [G[1]+4*G[2]]);

julia> is_pure(U)
true

julia> has_complement(U, G)[1]
true

is_neat(U::GrpAbFinGen, G::GrpAbFinGen) -> Bool

A subgroup U of G is called neat if for all primes p an element in U that is in the image of the multiplication by p map of G is also a multiple of an element in U.

See also: is_pure

EXAMPLES

julia> G = abelian_group([2, 8]);

julia> U, _ = sub(G, [G[1] + 2*G[2]]);

julia> is_neat(U, G)
true

julia> is_pure(U, G)
false

saturate(U::GrpAbFinGen, G::GrpAbFinGen) -> GrpAbFinGen

For a subgroup U of G find a minimal overgroup that is pure, and thus has a complement.

See also: is_pure, has_complement

psubgroups(g::GrpAbFinGen, p::Integer;
           subtype = :all,
           quotype = :all,
           index = -1,
           order = -1)

Return an iterator for the subgroups of $G$ of the specific form. Note that subtype (and quotype) is the type of the subgroup as an abelian $p$-group.

subgroups(g::GrpAbFinGen;
          subtype = :all ,
          quotype = :all,
          index = -1,
          order = -1)

Return an iterator for the subgroups of $G$ of the specific form.

quo(G::GrpAbFinGen, s::Vector{GrpAbFinGenElem}) -> GrpAbFinGen, GrpAbfinGemMap

Create the quotient $H$ of $G$ by the subgroup generated by the elements in $s$, together with the projection $p : G \to H$.

quo(G::GrpAbFinGen, M::ZZMatrix) -> GrpAbFinGen, GrpAbFinGenMap

Create the quotient $H$ of $G$ by the subgroup generated by the elements corresponding to the rows of $M$, together with the projection $p : G \to H$.

quo(G::GrpAbFinGen, n::Integer}) -> GrpAbFinGen, Map
quo(G::GrpAbFinGen, n::ZZRingElem}) -> GrpAbFinGen, Map

Returns the quotient $H = G/nG$ together with the projection $p : G \to H$.

quo(G::GrpAbFinGen, n::Integer}) -> GrpAbFinGen, Map
quo(G::GrpAbFinGen, n::ZZRingElem}) -> GrpAbFinGen, Map

Returns the quotient $H = G/nG$ together with the projection $p : G \to H$.

quo(G::GrpAbFinGen, U::GrpAbFinGen) -> GrpAbFinGen, Map

Create the quotient $H$ of $G$ by $U$, together with the projection $p : G \to H$.

intersect(mG::GrpAbFinGenMap, mH::GrpAbFinGenMap) -> GrpAbFinGen, Map

Given two injective maps of abelian groups with the same codomain $G$, return the intersection of the images as a subgroup of $G$.

direct_product(G::GrpAbFinGen...) -> GrpAbFinGen, Vector{GrpAbFinGenMap}

Return the direct product $D$ of the (finitely many) abelian groups $G_i$, together with the projections $D \to G_i$.

For finite abelian groups, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain $D$ as a direct sum together with the injections $D \to G_i$, one should call direct_sum(G...). If one wants to obtain $D$ as a biproduct together with the projections and the injections, one should call biproduct(G...).

Otherwise, one could also call canonical_injections(D) or canonical_projections(D) later on.

canonical_injection(G::GrpAbFinGen, i::Int) -> GrpAbFinGenMap

Given a group $G$ that was created as a direct product, return the injection from the $i$th component.

canonical_projection(G::GrpAbFinGen, i::Int) -> GrpAbFinGenMap

Given a group $G$ that was created as a direct product, return the projection onto the $i$th component.

flat(G::GrpAbFinGen) -> GrpAbFinGenMap

Given a group $G$ that is created using (iterated) direct products, or (iterated) tensor products, return a group isomorphism into a flat product: for $G := (A \oplus B) \oplus C$, it returns the isomorphism $G \to A \oplus B \oplus C$ (resp. $\otimes$).

tensor_product(G::GrpAbFinGen...; task::Symbol = :map) -> GrpAbFinGen, Map

Given groups $G_i$, compute the tensor product $G_1\otimes \cdots \otimes G_n$. If task is set to ":map", a map $\phi$ is returned that maps tuples in $G_1 \times \cdots \times G_n$ to pure tensors $g_1 \otimes \cdots \otimes g_n$. The map admits a preimage as well.

hom(G::GrpAbFinGen, H::GrpAbFinGen, A::Vector{ <: Map{GrpAbFinGen, GrpAbFinGen}}) -> Map

Given groups $G = G_1 \otimes \cdots \otimes G_n$ and $H = H_1 \otimes \cdot \otimes H_n$ as well as maps $\phi_i: G_i\to H_i$, compute the tensor product of the maps.

hom(G::GrpAbFinGen, H::GrpAbFinGen; task::Symbol = :map) -> GrpAbFinGen, Map

Computes the group of all homomorpisms from $G$ to $H$ as an abstract group. If task is ":map", then a map $\phi$ is computed that can be used to obtain actual homomorphisms. This map also allows preimages. Set task to ":none" to not compute the map.