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

# snf — Method.

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_isomorphism — Method.

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

# torsion_subgroup — Method.

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

Return the torsion subgroup of G.

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# sub — Method.

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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# sub — Method.

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 — Method.

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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# sub — Method.

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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# sub — Method.

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_subgroup — Method.

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_quotient — Method.

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_complement — Method.

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_pure — Method.

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

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# is_neat — Method.

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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# saturate — Method.

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

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

# psubgroups — Method.

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/12

julia> shapes = MSet{Vector{ZZRingElem}}()
MSet{Vector{ZZRingElem}}()

julia> for U = psubgroups(G, 2)
         push!(shapes, elementary_divisors(U[1]))
       end

julia> shapes
MSet{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.

# subgroups — Method.

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

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

source

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]])

# quo — Method.

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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# quo — Method.

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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# quo — Method.

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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# quo — Method.

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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# quo — Method.

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.

# intersect — Method.

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_product — Method.

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_injection — Method.

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_projection — Method.

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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# flat — Method.

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_product — Method.

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 $\varphi$ 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 — Method.

hom(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 ${\varphi }_i:G_i\to H_i$, compute the tensor product of the maps.

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Hom-Group

# hom — Method.

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 $\varphi$ 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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