Structural Computations

snf(A::FinGenAbGroup) -> FinGenAbGroup, FinGenAbGroupHom

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::FinGenAbGroup) -> FinGenAbGroup, Map

Return the torsion subgroup of G.

sub(G::FinGenAbGroup, s::Vector{FinGenAbGroupElem}) -> FinGenAbGroup, FinGenAbGroupHom

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

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

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

sub(s::Vector{FinGenAbGroupElem}) -> FinGenAbGroup, FinGenAbGroupHom

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(F::FreeMod{T}, V::Vector{<:FreeModElem{T}}; cache_morphism::Bool=false) where {T}

Given a vector V of (homogeneous) elements of F, return a pair (I, inc) consisting of the (graded) submodule I of F generated by these elements and its inclusion map inc : I ↪ F.

When cache_morphism is set to true, then inc will be cached and available for transport and friends.

If only the submodule itself is desired, use sub_object instead.

sub(F::FreeMod{T}, A::MatElem{T}; cache_morphism::Bool=false) where {T}

Given a (homogeneous) matrix A interpret the rows of A as elements of the free module F and return a pair (I, inc) consisting of the (graded) submodule I of F generated by these row vectors, together with its inclusion map inc : I ↪ F.

When cache_morphism is set to true, then inc will be cached and available for transport and friends.

If only the submodule itself is desired, use sub_object instead.

sub(F::FreeMod{T}, O::Vector{<:SubquoModuleElem{T}}; cache_morphism::Bool=false) where T

Suppose the ambient_free_module of the parent M of the elements v_i in O is F and M is a submodule (i.e. no relations are present). Then this returns a pair (I, inc) consisting of the submodule I generated by the elements in O in F, together with its inclusion morphism inc : I ↪ F.

When cache_morphism is set to true, then inc will be cached and available for transport and friends.

If only the submodule itself is desired, use sub_object instead.

sub(F::FreeMod{T}, M::SubquoModule{T}; cache_morphism::Bool=false) where T

Return M as a submodule of F, together with its inclusion morphism inc : M ↪ F.

When cache_morphism is set to true, then inc will be cached and available for transport and friends.

The ambient_free_module of M needs to be F and M has to have no relations.

If only the submodule itself is desired, use sub_object instead.

sub(M::SubquoModule{T}, V::Vector{<:SubquoModuleElem{T}}; cache_morphism::Bool=false) where T

Given a vector V of (homogeneous) elements of M, return the (graded) submodule I of M generated by these elements together with its inclusion map `inc : I ↪ M.

When cache_morphism is set to true, then inc will be cached and available for transport and friends.

If only the submodule itself is desired, use sub_object instead.

sub(M::ModuleFP{T}, V::Vector{<:ModuleFPElem{T}}; cache_morphism::Bool=false) where T

Given a vector V of (homogeneous) elements of M, return the (graded) submodule I of M generated by these elements together with its inclusion map `inc : I ↪ M.

When cache_morphism is set to true, then inc will be cached and available for transport and friends.

If only the submodule itself is desired, use sub_object instead.

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
Module homomorphism
  from N
  to F

sub(G::FinGenAbGroup, M::ZZMatrix) -> FinGenAbGroup, FinGenAbGroupHom

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::FinGenAbGroup, n::ZZRingElem) -> FinGenAbGroup, FinGenAbGroupHom

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

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

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

sylow_subgroup(G::FinGenAbGroup, p::IntegerUnion) -> FinGenAbGroup, FinGenAbGroupHom

Return the Sylow $p-$subgroup of the finitely generated abelian group G, for a prime p. This is the subgroup of p-power order in G whose index in G is coprime to p.

Examples

julia> A = abelian_group(ZZRingElem[2, 6, 30])
Z/2 x Z/6 x Z/30

julia> H, j = sylow_subgroup(A, 2);

julia> H
(Z/2)^3

julia> divexact(order(A), order(H))
45

has_quotient(G::FinGenAbGroup, 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::FinGenAbGroupHom) -> Bool, FinGenAbGroupHom
has_complement(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool, FinGenAbGroupHom

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::FinGenAbGroup, G::FinGenAbGroup) -> 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::FinGenAbGroup, G::FinGenAbGroup) -> 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::FinGenAbGroup, G::FinGenAbGroup) -> FinGenAbGroup

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::FinGenAbGroup, 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::FinGenAbGroup;
          subtype = :all ,
          quotype = :all,
          index = -1,
          order = -1)

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

quo(G::FinGenAbGroup, s::Vector{FinGenAbGroupElem}) -> FinGenAbGroup, 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::FinGenAbGroup, M::ZZMatrix) -> FinGenAbGroup, FinGenAbGroupHom

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

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

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

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

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

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

intersect(mG::FinGenAbGroupHom, mH::FinGenAbGroupHom) -> FinGenAbGroup, 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::FinGenAbGroup...) -> FinGenAbGroup, Vector{FinGenAbGroupHom}

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::FinGenAbGroup, i::Int) -> FinGenAbGroupHom

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

canonical_projection(G::FinGenAbGroup, i::Int) -> FinGenAbGroupHom

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

flat(G::FinGenAbGroup) -> FinGenAbGroupHom

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::FinGenAbGroup...; task::Symbol = :map) -> FinGenAbGroup, 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_tensor(G::FinGenAbGroup, H::FinGenAbGroup, A::Vector{ <: Map{FinGenAbGroup, FinGenAbGroup}}) -> 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::FinGenAbGroup, H::FinGenAbGroup; task::Symbol = :map) -> FinGenAbGroup, 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.