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.

the zero point for AbHom.

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 density of the abelian homomorphism, i. e. the density of the underlying matrix (see: mtxDensity).

splitable property for AbHom with free start point of any finite dimension.

the projection AbHomFree as left adjoint.

projection homomorphisms to Matrix Z.

products for AbHom.

sums for AbHom.

free finitely presentations for AbHom.

elements of an finitely generated abelian group. There root - which is an element of AbGroup - gives there affiliated group. They are gererated via make.

form for a AbElement.

the i-th canonical generator of the given abelian group.

the underlying Z-vector.

the abelian homomorphism with the free start point of dimension 1 and free end point of the given dimension according to the given vector.

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.

random variable for some free slice from in AbHom

validity of the algebraic structure of AbHom.