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# Groups

class `sage.categories.groups.``Groups`(base_category)

  Bases: `sage.categories.category_with_axiom.CategoryWithAxiom_singleton`

  The category of (multiplicative) groups, i.e. monoids with inverses.

  EXAMPLES:

      sage: Groups()
      Category of groups
      sage: Groups().super_categories()
      [Category of monoids, Category of inverse unital magmas]

  TESTS:

      sage: TestSuite(Groups()).run()

  class `Algebras`(category, *args)

    Bases: `sage.categories.algebra_functor.AlgebrasCategory`

    The category of group algebras over a given base ring.

    EXAMPLES:

        sage: GroupAlgebras(IntegerRing())
        Category of group algebras over Integer Ring
        sage: GroupAlgebras(IntegerRing()).super_categories()
        [Category of hopf algebras with basis over Integer Ring,
         Category of monoid algebras over Integer Ring]

    Here is how to create the group algebra of a group \(G\):

        sage: G = DihedralGroup(5)
        sage: QG = G.algebra(QQ); QG
        Group algebra of Dihedral group of order 10 as a permutation group over Rational Field

    and an example of computation:

        sage: g = G.an_element(); g
        (1,2,3,4,5)
        sage: (QG.term(g) + 1)**3
        B[()] + 3*B[(1,2,3,4,5)] + 3*B[(1,3,5,2,4)] + B[(1,4,2,5,3)]

    Todo

    - Check which methods would be better located in `Monoid.Algebras` or `Groups.Finite.Algebras`.

    TESTS:

        sage: A = GroupAlgebras(QQ).example(GL(3, GF(11)))
        sage: A.one_basis()
        [1 0 0]
        [0 1 0]
        [0 0 1]
        sage: A = SymmetricGroupAlgebra(QQ,4)
        sage: x = Permutation([4,3,2,1])
        sage: A.product_on_basis(x,x)
        [1, 2, 3, 4]

        sage: C = GroupAlgebras(ZZ)
        sage: TestSuite(C).run()

    class `ElementMethods`
      `central_form`()

        Return `self` in the canonical basis of the center of the group algebra.

        INPUT:

        - `self` – a central element of the group algebra

        OUTPUT:

        - A formal linear combination of the conjugacy class representatives representing its coordinates in the canonical basis of the center. See `Groups.Algebras.ParentMethods.center()` for details.

        Warning

        - This method requires the underlying group to have a method `conjugacy_classes_representatives` (every permutation group has one, thanks GAP!).
        - This method does not check that the element is indeed central. Use the method `Monoids.Algebras.ElementMethods.is_central()` for this purpose.
        - This function has a complexity linear in the number of conjugacy classes of the group. One could easily implement a function whose complexity is linear in the size of the support of `self`.

        EXAMPLES:

            sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
            sage: A=QS3([2,3,1])+QS3([3,1,2])
            sage: A.central_form()
            B[[2, 3, 1]]
            sage: QS4 = SymmetricGroupAlgebra(QQ, 4)
            sage: B=sum(len(s.cycle_type())*QS4(s) for s in Permutations(4))
            sage: B.central_form()
            4*B[[1, 2, 3, 4]] + 3*B[[2, 1, 3, 4]] + 2*B[[2, 1, 4, 3]] + 2*B[[2, 3, 1, 4]] + B[[2, 3, 4, 1]]
            sage: QG=GroupAlgebras(QQ).example(PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]))
            sage: sum(i for i in QG.basis()).central_form()
            B[()] + B[(4,5)] + B[(3,4,5)] + B[(2,3)(4,5)] + B[(2,3,4,5)] + B[(1,2)(3,4,5)] + B[(1,2,3,4,5)]

        See also

        - `Groups.Algebras.ParentMethods.center()`
        - `Monoids.Algebras.ElementMethods.is_central()`
    class `Groups.Algebras.``ParentMethods`
      `algebra_generators`()

        Return generators of this group algebra (as an algebra).

        EXAMPLES:

            sage: GroupAlgebras(QQ).example(AlternatingGroup(10)).algebra_generators()
            Finite family {(1,2,3,4,5,6,7,8,9): B[(1,2,3,4,5,6,7,8,9)], (8,9,10): B[(8,9,10)]}

        Note

        This function is overloaded for SymmetricGroupAlgebras to return Permutations and not Elements of the symmetric group:

            sage: GroupAlgebras(QQ).example(SymmetricGroup(10)).algebra_generators()
            [[2, 1, 3, 4, 5, 6, 7, 8, 9, 10], [2, 3, 4, 5, 6, 7, 8, 9, 10, 1]]

      `antipode_on_basis`(g)

        Return the antipode of the element `g` of the basis.

        Each basis element `g` is group-like, and so has antipode \(g^{-1}\). This method is used to compute the antipode of any element.

        EXAMPLES:

            sage: A=CyclicPermutationGroup(6).algebra(ZZ);A
            Group algebra of Cyclic group of order 6 as a permutation group over Integer Ring
            sage: g=CyclicPermutationGroup(6).an_element();g
            (1,2,3,4,5,6)
            sage: A.antipode_on_basis(g)
            B[(1,6,5,4,3,2)]
            sage: a=A.an_element();a
            B[()] + 3*B[(1,2,3,4,5,6)] + 3*B[(1,3,5)(2,4,6)]
            sage: a.antipode()
            B[()] + 3*B[(1,5,3)(2,6,4)] + 3*B[(1,6,5,4,3,2)]

      `center`()

        Return the center of the group algebra.

        The canonical basis of the center of the group algebra is the family \((f_\sigma)_{\sigma\in C}\), where \(C\) is any collection of representatives of the conjugacy classes of the group, and \(f_\sigma\) is the sum of the elements in the conjugacy class of \(\sigma\).

        OUTPUT:

        - A free module \(V\) indexed by conjugacy class representatives of the group; its elements represent formal linear combinations of the canonical basis elements.

        Warning

        - This method requires the underlying group to have a method `conjugacy_classes_representatives` (every permutation group has one, thanks GAP!).
        - The product has not been implemented yet.

        EXAMPLES:

            sage: SymmetricGroupAlgebra(ZZ,3).center()
            Free module generated by {[2, 3, 1], [2, 1, 3], [1, 2, 3]} over Integer Ring

        See also

        - `Groups.Algebras.ElementMethods.central_form()`
        - `Monoids.Algebras.ElementMethods.is_central()`
      `coproduct_on_basis`(g)

        Return the coproduct of the element `g` of the basis.

        Each basis element `g` is group-like. This method is used to compute the coproduct of any element.

        EXAMPLES:

            sage: A=CyclicPermutationGroup(6).algebra(ZZ);A
            Group algebra of Cyclic group of order 6 as a permutation group over Integer Ring
            sage: g=CyclicPermutationGroup(6).an_element();g
            (1,2,3,4,5,6)
            sage: A.coproduct_on_basis(g)
            B[(1,2,3,4,5,6)] # B[(1,2,3,4,5,6)]
            sage: a=A.an_element();a
            B[()] + 3*B[(1,2,3,4,5,6)] + 3*B[(1,3,5)(2,4,6)]
            sage: a.coproduct()
            B[()] # B[()] + 3*B[(1,2,3,4,5,6)] # B[(1,2,3,4,5,6)] + 3*B[(1,3,5)(2,4,6)] # B[(1,3,5)(2,4,6)]

      `counit`(x)

        Return the counit of the element `x` of the group algebra.

        This is the sum of all coefficients of `x` with respect to the standard basis of the group algebra.

        EXAMPLES:

            sage: A=CyclicPermutationGroup(6).algebra(ZZ);A
            Group algebra of Cyclic group of order 6 as a permutation group over Integer Ring
            sage: a=A.an_element();a
            B[()] + 3*B[(1,2,3,4,5,6)] + 3*B[(1,3,5)(2,4,6)]
            sage: a.counit()
            7

      `counit_on_basis`(g)

        Return the counit of the element `g` of the basis.

        Each basis element `g` is group-like, and so has counit \(1\). This method is used to compute the counit of any element.

        EXAMPLES:

            sage: A=CyclicPermutationGroup(6).algebra(ZZ);A
            Group algebra of Cyclic group of order 6 as a permutation group over Integer Ring
            sage: g=CyclicPermutationGroup(6).an_element();g
            (1,2,3,4,5,6)
            sage: A.counit_on_basis(g)
            1

      `group`()

        Return the underlying group of the group algebra.

        EXAMPLES:

            sage: GroupAlgebras(QQ).example(GL(3, GF(11))).group()
            General Linear Group of degree 3 over Finite Field of size 11
            sage: SymmetricGroupAlgebra(QQ,10).group()
            Symmetric group of order 10! as a permutation group

    `Groups.Algebras.``example`(G=None)

      Return an example of group algebra.

      EXAMPLES:

          sage: GroupAlgebras(QQ[x]).example()
          Group algebra of Dihedral group of order 8 as a permutation group over Univariate Polynomial Ring in x over Rational Field

      An other group can be specified as optional argument:

          sage: GroupAlgebras(QQ).example(AlternatingGroup(4))
          Group algebra of Alternating group of order 4!/2 as a permutation group over Rational Field

    `Groups.Algebras.``extra_super_categories`()

      Implement the fact that the algebra of a group is a Hopf algebra.

      EXAMPLES:

          sage: C = Groups().Algebras(QQ)
          sage: C.extra_super_categories()
          [Category of hopf algebras over Rational Field]
          sage: sorted(C.super_categories(), key=str)
          [Category of hopf algebras with basis over Rational Field,
           Category of monoid algebras over Rational Field]

  class `Groups.``CartesianProducts`(category, *args)

    Bases: `sage.categories.cartesian_product.CartesianProductsCategory`

    The category of groups constructed as cartesian products of groups.

    This construction gives the direct product of groups. See Wikipedia article Direct_product and Wikipedia article Direct_product_of_groups for more information.

    class `ParentMethods`
      `group_generators`()

        Return the group generators of `self`.

        EXAMPLES:

            sage: C5 = CyclicPermutationGroup(5)
            sage: C4 = CyclicPermutationGroup(4)
            sage: S4 = SymmetricGroup(3)
            sage: C = cartesian_product([C5, C4, S4])
            sage: C.group_generators()
            Family (((1,2,3,4,5), (), ()),
                    ((), (1,2,3,4), ()),
                    ((), (), (1,2)),
                    ((), (), (2,3)))

        We check the other portion of trac ticket #16718 is fixed:

            sage: len(C.j_classes())
            1

        An example with an infinitely generated group (a better output is needed):

            sage: G = Groups.free([1,2])
            sage: H = Groups.free(ZZ)
            sage: C = cartesian_product([G, H])
            sage: C.monoid_generators()
            Lazy family (gen(i))_{i in The cartesian product of (...)}

    `Groups.CartesianProducts.``extra_super_categories`()

      A cartesian product of groups is endowed with a natural group structure.

      EXAMPLES:

          sage: C = Groups().CartesianProducts()
          sage: C.extra_super_categories()
          [Category of groups]
          sage: sorted(C.super_categories(), key=str)
          [Category of Cartesian products of inverse unital magmas,
           Category of Cartesian products of monoids,
           Category of groups]

  class `Groups.``Commutative`(base_category)

    Bases: `sage.categories.category_with_axiom.CategoryWithAxiom`

    Category of commutative (abelian) groups.

    A group \(G\) is commutative if \(xy = yx\) for all \(x,y \in G\).

    static `free`(index_set=None, names=None, **kwds)

      Return the free commutative group.

      INPUT:

      - `index_set` – (optional) an index set for the generators; if an integer, then this represents \(\{0, 1, \ldots, n-1\}\)
      - `names` – a string or list/tuple/iterable of strings (default: `'x'`); the generator names or name prefix

      EXAMPLES:

          sage: Groups.Commutative.free(index_set=ZZ)
          Free abelian group indexed by Integer Ring
          sage: Groups().Commutative().free(ZZ)
          Free abelian group indexed by Integer Ring
          sage: Groups().Commutative().free(5)
          Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z
          sage: F.<x,y,z> = Groups().Commutative().free(); F
          Multiplicative Abelian group isomorphic to Z x Z x Z

  class `Groups.``ElementMethods`
  `Groups.``Finite`

    alias of `FiniteGroups`

  class `Groups.``ParentMethods`
    `cayley_table`(names='letters', elements=None)

      Returns the “multiplication” table of this multiplicative group, which is also known as the “Cayley table”.

      Note

      The order of the elements in the row and column headings is equal to the order given by the table’s `column_keys()` method. The association between the actual elements and the names/symbols used in the table can also be retrieved as a dictionary with the `translation()` method.

      For groups, this routine should behave identically to the `multiplication_table()` method for magmas, which applies in greater generality.

      INPUT:

      - `names` - the type of names used, values are:
        - `'letters'` - lowercase ASCII letters are used for a base 26 representation of the elements’ positions in the list given by `list()`, padded to a common width with leading ‘a’s.
        - `'digits'` - base 10 representation of the elements’ positions in the list given by `column_keys()`, padded to a common width with leading zeros.
        - `'elements'` - the string representations of the elements themselves.
        - a list - a list of strings, where the length of the list equals the number of elements.
      - `elements` - default = `None`. A list of elements of the group, in forms that can be coerced into the structure, eg. their string representations. This may be used to impose an alternate ordering on the elements, perhaps when this is used in the context of a particular structure. The default is to use whatever ordering is provided by the the group, which is reported by the `column_keys()` method. Or the `elements` can be a subset which is closed under the operation. In particular, this can be used when the base set is infinite.

      OUTPUT: An object representing the multiplication table. This is an `OperationTable` object and even more documentation can be found there.

      EXAMPLES:

      Permutation groups, matrix groups and abelian groups can all compute their multiplication tables.

          sage: G = DiCyclicGroup(3)
          sage: T = G.cayley_table()
          sage: T.column_keys()
          ((), (5,6,7), (5,7,6)...(1,4,2,3)(5,7))
          sage: T
          *  a b c d e f g h i j k l
           +------------------------
          a| a b c d e f g h i j k l
          b| b c a e f d i g h l j k
          c| c a b f d e h i g k l j
          d| d e f a b c j k l g h i
          e| e f d b c a l j k i g h
          f| f d e c a b k l j h i g
          g| g h i j k l d e f a b c
          h| h i g k l j f d e c a b
          i| i g h l j k e f d b c a
          j| j k l g h i a b c d e f
          k| k l j h i g c a b f d e
          l| l j k i g h b c a e f d

          sage: M=SL(2,2)
          sage: M.cayley_table()
          *  a b c d e f
           +------------
          a| a b c d e f
          b| b a d c f e
          c| c f e b a d
          d| d e f a b c
          e| e d a f c b
          f| f c b e d a
          <BLANKLINE>

          sage: A=AbelianGroup([2,3])
          sage: A.cayley_table()
          *  a b c d e f
           +------------
          a| a b c d e f
          b| b c a e f d
          c| c a b f d e
          d| d e f a b c
          e| e f d b c a
          f| f d e c a b

      Lowercase ASCII letters are the default symbols used for the table, but you can also specify the use of decimal digit strings, or provide your own strings (in the proper order if they have meaning). Also, if the elements themselves are not too complex, you can choose to just use the string representations of the elements themselves.

          sage: C=CyclicPermutationGroup(11)
          sage: C.cayley_table(names='digits')
           *  00 01 02 03 04 05 06 07 08 09 10
            +---------------------------------
          00| 00 01 02 03 04 05 06 07 08 09 10
          01| 01 02 03 04 05 06 07 08 09 10 00
          02| 02 03 04 05 06 07 08 09 10 00 01
          03| 03 04 05 06 07 08 09 10 00 01 02
          04| 04 05 06 07 08 09 10 00 01 02 03
          05| 05 06 07 08 09 10 00 01 02 03 04
          06| 06 07 08 09 10 00 01 02 03 04 05
          07| 07 08 09 10 00 01 02 03 04 05 06
          08| 08 09 10 00 01 02 03 04 05 06 07
          09| 09 10 00 01 02 03 04 05 06 07 08
          10| 10 00 01 02 03 04 05 06 07 08 09

          sage: G=QuaternionGroup()
          sage: names=['1', 'I', '-1', '-I', 'J', '-K', '-J', 'K']
          sage: G.cayley_table(names=names)
           *   1  I -1 -I  J -K -J  K
            +------------------------
           1|  1  I -1 -I  J -K -J  K
           I|  I -1 -I  1  K  J -K -J
          -1| -1 -I  1  I -J  K  J -K
          -I| -I  1  I -1 -K -J  K  J
           J|  J -K -J  K -1 -I  1  I
          -K| -K -J  K  J  I -1 -I  1
          -J| -J  K  J -K  1  I -1 -I
           K|  K  J -K -J -I  1  I -1

          sage: A=AbelianGroup([2,2])
          sage: A.cayley_table(names='elements')
              *      1    f1    f0 f0*f1
               +------------------------
              1|     1    f1    f0 f0*f1
             f1|    f1     1 f0*f1    f0
             f0|    f0 f0*f1     1    f1
          f0*f1| f0*f1    f0    f1     1

      The `change_names()` routine behaves similarly, but changes an existing table “in-place.”

          sage: G=AlternatingGroup(3)
          sage: T=G.cayley_table()
          sage: T.change_names('digits')
          sage: T
          *  0 1 2
           +------
          0| 0 1 2
          1| 1 2 0
          2| 2 0 1

      For an infinite group, you can still work with finite sets of elements, provided the set is closed under multiplication. Elements will be coerced into the group as part of setting up the table.

          sage: G=SL(2,ZZ)
          sage: G
          Special Linear Group of degree 2 over Integer Ring
          sage: identity = matrix(ZZ, [[1,0], [0,1]])
          sage: G.cayley_table(elements=[identity, -identity])
          *  a b
           +----
          a| a b
          b| b a

      The `OperationTable` class provides even greater flexibility, including changing the operation. Here is one such example, illustrating the computation of commutators. `commutator` is defined as a function of two variables, before being used to build the table. From this, the commutator subgroup seems obvious, and creating a Cayley table with just these three elements confirms that they form a closed subset in the group.

          sage: from sage.matrix.operation_table import OperationTable
          sage: G=DiCyclicGroup(3)
          sage: commutator = lambda x, y: x*y*x^-1*y^-1
          sage: T=OperationTable(G, commutator)
          sage: T
          .  a b c d e f g h i j k l
           +------------------------
          a| a a a a a a a a a a a a
          b| a a a a a a c c c c c c
          c| a a a a a a b b b b b b
          d| a a a a a a a a a a a a
          e| a a a a a a c c c c c c
          f| a a a a a a b b b b b b
          g| a b c a b c a c b a c b
          h| a b c a b c b a c b a c
          i| a b c a b c c b a c b a
          j| a b c a b c a c b a c b
          k| a b c a b c b a c b a c
          l| a b c a b c c b a c b a
          sage: trans = T.translation()
          sage: comm = [trans['a'], trans['b'],trans['c']]
          sage: comm
          [(), (5,6,7), (5,7,6)]
          sage: P=G.cayley_table(elements=comm)
          sage: P
          *  a b c
           +------
          a| a b c
          b| b c a
          c| c a b

      TODO:

      Arrange an ordering of elements into cosets of a normal subgroup close to size \(\sqrt{n}\). Then the quotient group structure is often apparent in the table. See comments on Trac #7555.

      AUTHOR:

      - Rob Beezer (2010-03-15)
    `group_generators`()

      Returns group generators for self.

      This default implementation calls `gens()`, for backward compatibility.

      EXAMPLES:

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

    `holomorph`()

      The holomorph of a group

      The holomorph of a group \(G\) is the semidirect product \(G \rtimes_{id} Aut(G)\), where \(id\) is the identity function on \(Aut(G)\), the automorphism group of \(G\).

      See Wikipedia article Holomorph (mathematics)

      EXAMPLES:

          sage: G = Groups().example()
          sage: G.holomorph()
          Traceback (most recent call last):
          ...
          NotImplementedError: holomorph of General Linear Group of degree 4 over Rational Field not yet implemented

    `monoid_generators`()

      Return the generators of `self` as a monoid.

      Let \(G\) be a group with generating set \(X\). In general, the generating set of \(G\) as a monoid is given by \(X \cup X^{-1}\), where \(X^{-1}\) is the set of inverses of \(X\). If \(G\) is a finite group, then the generating set as a monoid is \(X\).

      EXAMPLES:

          sage: A = AlternatingGroup(4)
          sage: A.monoid_generators()
          Family ((2,3,4), (1,2,3))
          sage: F.<x,y> = FreeGroup()
          sage: F.monoid_generators()
          Family (x, y, x^-1, y^-1)

    `semidirect_product`(N, mapping, check=True)

      The semi-direct product of two groups

      EXAMPLES:

          sage: G = Groups().example()
          sage: G.semidirect_product(G,Morphism(G,G))
          Traceback (most recent call last):
          ...
          NotImplementedError: semidirect product of General Linear Group of degree 4 over Rational Field and General Linear Group of degree 4 over Rational Field not yet implemented

  `Groups.``example`()

    EXAMPLES:

        sage: Groups().example()
        General Linear Group of degree 4 over Rational Field

  static `Groups.``free`(index_set=None, names=None, **kwds)

    Return the free group.

    INPUT:

    - `index_set` – (optional) an index set for the generators; if an integer, then this represents \(\{0, 1, \ldots, n-1\}\)
    - `names` – a string or list/tuple/iterable of strings (default: `'x'`); the generator names or name prefix

    When the index set is an integer or only variable names are given, this returns `FreeGroup_class`, which currently has more features due to the interface with GAP than `IndexedFreeGroup`.

    EXAMPLES:

        sage: Groups.free(index_set=ZZ)
        Free group indexed by Integer Ring
        sage: Groups().free(ZZ)
        Free group indexed by Integer Ring
        sage: Groups().free(5)
        Free Group on generators {x0, x1, x2, x3, x4}
        sage: F.<x,y,z> = Groups().free(); F
        Free Group on generators {x, y, z}

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