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)]
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
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
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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