### Navigation

- index
- modules |
- next |
- previous |
- Category Theory »

# FiniteGroups

class `sage.categories.finite_groups.``FiniteGroups`(base_category)

  Bases: `sage.categories.category_with_axiom.CategoryWithAxiom`

  The category of finite (multiplicative) groups.

  EXAMPLES:

      sage: C = FiniteGroups(); C
      Category of finite groups
      sage: C.super_categories()
      [Category of finite monoids, Category of groups]
      sage: C.example()
      General Linear Group of degree 2 over Finite Field of size 3

  TESTS:

      sage: TestSuite(C).run()

  class `ElementMethods`
    `conjugacy_class`()

      Return the conjugacy class of `self`.

      EXAMPLES:

          sage: H = MatrixGroup([matrix(GF(5),2,[1,2, -1, 1]), matrix(GF(5),2, [1,1, 0,1])])
          sage: h = H(matrix(GF(5),2,[1,2, -1, 1]))
          sage: h.conjugacy_class()
          Conjugacy class of [1 2]
          [4 1] in Matrix group over Finite Field of size 5 with 2 generators (
          [1 2]  [1 1]
          [4 1], [0 1]
          )

  class `FiniteGroups.``ParentMethods`
    `cardinality`()

      Returns the cardinality of `self`, as per `EnumeratedSets.ParentMethods.cardinality()`.

      This default implementation calls `order()` if available, and otherwise resorts to `_cardinality_from_iterator()`. This is for backward compatibility only. Finite groups should override this method instead of `order()`.

      EXAMPLES:

      We need to use a finite group which uses this default implementation of cardinality:

          sage: R.<x> = PolynomialRing(QQ)
          sage: f = x^4 - 17*x^3 - 2*x + 1
          sage: G = f.galois_group(pari_group=True); G
          PARI group [24, -1, 5, "S4"] of degree 4
          sage: G.cardinality.__module__
          'sage.categories.finite_groups'
          sage: G.cardinality()
          24

    `cayley_graph_disabled`(connecting_set=None)

      AUTHORS:

      - Bobby Moretti (2007-08-10)
      - Robert Miller (2008-05-01): editing
    `conjugacy_class`(g)

      Return the conjugacy class of the element `g`.

      This is a fall-back method for groups not defined over GAP.

      EXAMPLES:

          sage: W = WeylGroup(['C',6])
          sage: c = W.conjugacy_class(W.an_element())
          sage: type(c)
          <class 'sage.groups.conjugacy_classes.ConjugacyClass_with_category'>

    `conjugacy_classes`()

      Return a list with all the conjugacy classes of the group.

      This will eventually be a fall-back method for groups not defined over GAP. Right now just raises a `NotImplementedError`, until we include a non-GAP way of listing the conjugacy classes representatives.

      EXAMPLES:

          sage: from sage.groups.group import FiniteGroup
          sage: G = FiniteGroup()
          sage: G.conjugacy_classes()
          Traceback (most recent call last):
          ...
          NotImplementedError: Listing the conjugacy classes for
          group <type 'sage.groups.group.FiniteGroup'> is not implemented

    `conjugacy_classes_representatives`()

      Return a list of the conjugacy classes representatives of the group.

      EXAMPLES:

          sage: G = SymmetricGroup(3)
          sage: G.conjugacy_classes_representatives()
          [(), (1,2), (1,2,3)]

    `monoid_generators`()

      Return monoid generators for `self`.

      For finite groups, the group generators are also monoid generators. Hence, this default implementation calls `group_generators()`.

      EXAMPLES:

          sage: A = AlternatingGroup(4)
          sage: A.monoid_generators()
          Family ((2,3,4), (1,2,3))

    `semigroup_generators`()

      Returns semigroup generators for self.

      For finite groups, the group generators are also semigroup generators. Hence, this default implementation calls `group_generators()`.

      EXAMPLES:

          sage: A = AlternatingGroup(4)
          sage: A.semigroup_generators()
          Family ((2,3,4), (1,2,3))

    `some_elements`()

      Return some elements of `self`.

      EXAMPLES:

          sage: A = AlternatingGroup(4)
          sage: A.some_elements()
          [(2,3,4), (1,2,3)]

  `FiniteGroups.``example`()

    Return an example of finite group, as per `Category.example()`.

    EXAMPLES:

        sage: G = FiniteGroups().example(); G
        General Linear Group of degree 2 over Finite Field of size 3

### Navigation

- index
- modules |
- next |
- previous |
- Category Theory »