## Structural Computations

Abelian groups support a wide range of structural operations such as

• enumeration of subgroups
• (outer) direct products
• tensor and hom constructions
• free resolutions and general complexes
• (co)-homology and tensor and hom-functors
snfMethod
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 $f : G \to A$.

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find_isomorphismMethod
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.

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### Subgroups and Quotients

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

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

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

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

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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 $\iota : H \to G$.

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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.

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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.

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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.

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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.

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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.

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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
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 R
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subMethod
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 $\iota : H \to G$.

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subMethod
sub(G::FinGenAbGroup, n::ZZRingElem) -> FinGenAbGroup, FinGenAbGroupHom

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

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

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

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sylow_subgroupMethod
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
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has_quotientMethod
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.

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has_complementMethod
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

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is_pureMethod
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.

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
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is_neatMethod
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
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saturateMethod
saturate(U::FinGenAbGroup, G::FinGenAbGroup) -> FinGenAbGroup

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

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A sophisticated algorithm for the enumeration of all (or selected) subgroups of a finite abelian group is available.

psubgroupsMethod
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.

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julia> G = abelian_group([6, 12])Z/6 x Z/12julia> shapes = MSet{Vector{ZZRingElem}}()MSet{Vector{ZZRingElem}}()julia> for U = psubgroups(G, 2)
push!(shapes, elementary_divisors(U[1]))
endjulia> shapesMSet{Vector{ZZRingElem}} with 8 elements:
ZZRingElem[]
ZZRingElem[4]    : 2
ZZRingElem[2, 4]
ZZRingElem[2]    : 3
ZZRingElem[2, 2]

So there are $2$ subgroups isomorphic to $C_4$ (ZZRingElem[4] : 2), $1$ isomorphic to $C_2\times C_4$, 1 trivial and $3$ $C_2$ subgroups.

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

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

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julia> for U = subgroups(G, subtype = [2])
@show U[1], map(U[2], gens(U[1]))
end(U[1], map(U[2], gens(U[1]))) = (Z/2, FinGenAbGroupElem[[0, 6]])
(U[1], map(U[2], gens(U[1]))) = (Z/2, FinGenAbGroupElem[[3, 6]])
(U[1], map(U[2], gens(U[1]))) = (Z/2, FinGenAbGroupElem[[3, 0]])julia> for U = subgroups(G, quotype = [2])
@show U[1], map(U[2], gens(U[1]))
end(U[1], map(U[2], gens(U[1]))) = (Finitely generated abelian group with 3 generators and 3 relations, FinGenAbGroupElem[[3, 3], [0, 4], [2, 0]])
(U[1], map(U[2], gens(U[1]))) = (Finitely generated abelian group with 3 generators and 3 relations, FinGenAbGroupElem[[0, 3], [0, 4], [2, 0]])
(U[1], map(U[2], gens(U[1]))) = (Finitely generated abelian group with 4 generators and 4 relations, FinGenAbGroupElem[[3, 6], [0, 6], [0, 4], [2, 0]])
quoMethod
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 $p : G \to H$.

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quoMethod
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 $p : G \to H$.

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

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

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

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

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

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

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For 2 subgroups U and V of the same group G, U+V returns the smallest subgroup of G containing both. Similarly, $U\cap V$ computes the intersection and $U \subset V$ tests for inclusion. The difference between issubset = $\subset$ and is_subgroup is that the inclusion map is also returned in the 2nd call.

intersectMethod
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 $G$.

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### Direct Products

direct_productMethod
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.

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canonical_injectionMethod
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.

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canonical_projectionMethod
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.

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flatMethod
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. $\otimes$).

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### Tensor Producs

tensor_productMethod
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.

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hom_tensorMethod
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.

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### Hom-Group

homMethod
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.

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