Finite groups

class sage.categories.finite_groups.FiniteGroups(base_category)[source]

Bases: CategoryWithAxiom_singleton

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

>>> from sage.all import *
>>> C = FiniteGroups(); C
Category of finite groups
>>> C.super_categories()
[Category of finite monoids, Category of groups]
>>> C.example()
General Linear Group of degree 2 over Finite Field of size 3
class Algebras(category, *args)[source]

Bases: AlgebrasCategory

class ParentMethods[source]

Bases: object

extra_super_categories()[source]

Implement Maschke’s theorem.

In characteristic 0 all finite group algebras are semisimple.

EXAMPLES:

sage: FiniteGroups().Algebras(QQ).is_subcategory(Algebras(QQ).Semisimple())
True
sage: FiniteGroups().Algebras(FiniteField(7)).is_subcategory(Algebras(FiniteField(7)).Semisimple())
False
sage: FiniteGroups().Algebras(ZZ).is_subcategory(Algebras(ZZ).Semisimple())
False
sage: FiniteGroups().Algebras(Fields()).is_subcategory(Algebras(Fields()).Semisimple())
False

sage: Cat = CommutativeAdditiveGroups().Finite()
sage: Cat.Algebras(QQ).is_subcategory(Algebras(QQ).Semisimple())
True
sage: Cat.Algebras(GF(7)).is_subcategory(Algebras(GF(7)).Semisimple())
False
sage: Cat.Algebras(ZZ).is_subcategory(Algebras(ZZ).Semisimple())
False
sage: Cat.Algebras(Fields()).is_subcategory(Algebras(Fields()).Semisimple())
False

>>> from sage.all import *
>>> FiniteGroups().Algebras(QQ).is_subcategory(Algebras(QQ).Semisimple())
True
>>> FiniteGroups().Algebras(FiniteField(Integer(7))).is_subcategory(Algebras(FiniteField(Integer(7))).Semisimple())
False
>>> FiniteGroups().Algebras(ZZ).is_subcategory(Algebras(ZZ).Semisimple())
False
>>> FiniteGroups().Algebras(Fields()).is_subcategory(Algebras(Fields()).Semisimple())
False

>>> Cat = CommutativeAdditiveGroups().Finite()
>>> Cat.Algebras(QQ).is_subcategory(Algebras(QQ).Semisimple())
True
>>> Cat.Algebras(GF(Integer(7))).is_subcategory(Algebras(GF(Integer(7))).Semisimple())
False
>>> Cat.Algebras(ZZ).is_subcategory(Algebras(ZZ).Semisimple())
False
>>> Cat.Algebras(Fields()).is_subcategory(Algebras(Fields()).Semisimple())
False
class ElementMethods[source]

Bases: object

class ParentMethods[source]

Bases: object

cardinality()[source]

Return 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: G = groups.misc.SemimonomialTransformation(GF(5), 3); G           # needs sage.rings.number_field
Semimonomial transformation group over Finite Field of size 5 of degree 3
sage: G.cardinality.__module__                                          # needs sage.rings.number_field
'sage.categories.finite_groups'
sage: G.cardinality()                                                   # needs sage.rings.number_field
384

>>> from sage.all import *
>>> G = groups.misc.SemimonomialTransformation(GF(Integer(5)), Integer(3)); G           # needs sage.rings.number_field
Semimonomial transformation group over Finite Field of size 5 of degree 3
>>> G.cardinality.__module__                                          # needs sage.rings.number_field
'sage.categories.finite_groups'
>>> G.cardinality()                                                   # needs sage.rings.number_field
384
cayley_graph_disabled(connecting_set=None)[source]

AUTHORS:

  • Bobby Moretti (2007-08-10)

  • Robert Miller (2008-05-01): editing

conjugacy_classes()[source]

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, it 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 <sage.groups.group.FiniteGroup object at ...> is not implemented

>>> from sage.all import *
>>> from sage.groups.group import FiniteGroup
>>> G = FiniteGroup()
>>> G.conjugacy_classes()
Traceback (most recent call last):
...
NotImplementedError: Listing the conjugacy classes for group <sage.groups.group.FiniteGroup object at ...> is not implemented
conjugacy_classes_representatives()[source]

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

EXAMPLES:

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

>>> from sage.all import *
>>> G = SymmetricGroup(Integer(3))
>>> G.conjugacy_classes_representatives()                             # needs sage.combinat
[(), (1,2), (1,2,3)]
monoid_generators()[source]

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 ((1,2,3), (2,3,4))

>>> from sage.all import *
>>> A = AlternatingGroup(Integer(4))
>>> A.monoid_generators()
Family ((1,2,3), (2,3,4))
semigroup_generators()[source]

Return 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 ((1,2,3), (2,3,4))

>>> from sage.all import *
>>> A = AlternatingGroup(Integer(4))
>>> A.semigroup_generators()
Family ((1,2,3), (2,3,4))
some_elements()[source]

Return some elements of self.

EXAMPLES:

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

>>> from sage.all import *
>>> A = AlternatingGroup(Integer(4))
>>> A.some_elements()
Family ((1,2,3), (2,3,4))
example()[source]

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

>>> from sage.all import *
>>> G = FiniteGroups().example(); G
General Linear Group of degree 2 over Finite Field of size 3