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# Homology Groups

This module defines a `HomologyGroup()` class which is an abelian group that prints itself in a way that is suitable for homology groups.

`sage.homology.homology_group.``HomologyGroup`(n, base_ring, invfac=None)

  Abelian group on \(n\) generators which represents a homology group in a fixed degree.

  INPUT:

  - `n` – integer; the number of generators
  - `base_ring` – ring; the base ring over which the homology is computed
  - `inv_fac` – list of integers; the invariant factors – ignored if the base ring is a field

  OUTPUT:

  A class that can represent the homology group in a fixed homological degree.

  EXAMPLES:

      sage: from sage.homology.homology_group import HomologyGroup
      sage: G = AbelianGroup(5, [5,5,7,8,9]); G
      Multiplicative Abelian group isomorphic to C5 x C5 x C7 x C8 x C9
      sage: H = HomologyGroup(5, ZZ, [5,5,7,8,9]); H
      C5 x C5 x C7 x C8 x C9
      sage: AbelianGroup(4)
      Multiplicative Abelian group isomorphic to Z x Z x Z x Z
      sage: HomologyGroup(4, ZZ)
      Z x Z x Z x Z
      sage: HomologyGroup(100, ZZ)
      Z^100

class `sage.homology.homology_group.``HomologyGroup_class`(n, invfac)

  Bases: `sage.groups.additive_abelian.additive_abelian_group.AdditiveAbelianGroup_fixed_gens`

  Discrete Abelian group on \(n\) generators. This class inherits from `AdditiveAbelianGroup_fixed_gens`; see `sage.groups.additive_abelian.additive_abelian_group` for more documentation. The main difference between the classes is in the print representation.

  EXAMPLES:

      sage: from sage.homology.homology_group import HomologyGroup
      sage: G = AbelianGroup(5, [5,5,7,8,9]); G
      Multiplicative Abelian group isomorphic to C5 x C5 x C7 x C8 x C9
      sage: H = HomologyGroup(5, ZZ, [5,5,7,8,9]); H
      C5 x C5 x C7 x C8 x C9
      sage: G == loads(dumps(G))
      True
      sage: AbelianGroup(4)
      Multiplicative Abelian group isomorphic to Z x Z x Z x Z
      sage: HomologyGroup(4, ZZ)
      Z x Z x Z x Z
      sage: HomologyGroup(100, ZZ)
      Z^100

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