finitely generate abelian group, i.e. the cartesian product of cyclic groups Z/n and are represented as a formal product with symbols in ZMod.

Definition Let g be in AbGroup. We call g smith normal if and only if there exists a sequence n 0, n 1 .. n (k-1) in N with length k and a exponent r in N such that:

1. 2 <= n i for all 0 <= i < k.
2. n (i + 1) mod n i == 0 for all i in 0 <= i < k-1.
3. g == abg (n 0) * abg (n 1) * .. * abg (n (k-1)) * abg 0 ^ r.

Theorem Every finitely generated abelian group is isomorphic to a group in smith normal form. This isomorphism is given by isoSmithNormal.

Examples Finitely generated abelian groups constructed via abg and its multiplicative structure:

>>> abg 12
AbGroup[Z/12]

represents the cyclic group Z/12.

>>> abg 2 * abg 3
AbGroup[Z/2*Z/3]

represents the cartesian product of the groups Z/2 and Z/3.

>>> abg 6 * abg 4 * abg 4
AbGroup[Z/6*Z/4^2]

represents the cartesian product of the groups Z/6, Z/4 and Z/4.

>>> abg 0 ^ 6
AbGroup[Z^6]

represents the free abelian group Z ^ 6 of dimension 6.

>>> one () :: AbGroup
AbGroup[]

represents the cartesian product of zero cyclic groups and

>>> one () * abg 4 * abg 6 == abg 4 * abg 6
True

Examples Checking for smith normal via isSmithNormal:

>>> isSmithNormal (abg 4)
True

>>> isSmithNormal (abg 2 * abg 2)
True

>>> isSmithNormal (abg 17 * abg 51)
True

>>> isSmithNormal (abg 2 * abg 4 * abg 0 ^ 3)
True

>>> isSmithNormal (abg 5 * abg 3)
False

>>> isSmithNormal (abg 0 * abg 3 * abg 6)
False

>>> isSmithNormal (abg 1 * abg 4)
False

>>> isSmithNormal (one ())
True

Examples The associated isomorphism in AbHom of a finitely generated abelian group given by isoSmithNormal.

>>> end (isoSmithNormal (abg 3 * abg 5))
AbGroup[Z/15]

>>> end (isoSmithNormal (abg 2 * abg 4 * abg 2))
AbGroup[Z/2^2*Z/4]

>>> end (isoSmithNormal (abg 4 * abg 6))
AbGroup[Z/2*Z/12]

>>> end (isoSmithNormal (abg 1))
AbGroup[]

the cyclic group of the given order as a finitely generated abelian group.

checks if the given group is smith normal (see definition AbGroup).

the associated dimension for matrices of ZModHom.

additive homomorphism between finitely generated abelian groups which are represented by matrices over ZModHom.

the additive homomorphism with the given orientation and ZModHom-entries.

the additive homomorphism with the given orientation and Z-entries.

the underlying Z-matrix.

the associated homomorphism between products of abg 0 given by the column - respectively row - length.

the projection AbHomFree as left adjoint.

projection homomorphisms to Matrix Z.

products for AbHom.

sums for AbHom.

the generator for a finitely generated abelian group.

Property Let a be in AbGroup, then holds a == g where Generator (DiagramChainTo g _) _ _ _ _ = abgGeneratorTo a.

random variable for AbHom given by a density and an orientation.

random variable of homomorphisms between abelian groups with end equal to the given one.

   r s t
  [f    ] a
  [     ] b
  [g h  ] c

random variable of homomorphisms between abelian groups with start equal to the given one.

   a b c
  [f    ] r
  [g   l] s
  [h    ] t

the maximal length of abelian groups for the standard random variable of type X AbGroup.

Property 1 <= stdMaxDim.

validity of the algebraic structure of AbHom.