An additive group is a set with an internal binary operation $$+$$ which is associative, admits a zero, and where every element can be negated.

EXAMPLES:

sage: from sage.categories.additive_groups import AdditiveGroups
Category of sets,
Category of sets with partial maps,
Category of objects]

True

>>> from sage.all import *
Category of sets,
Category of sets with partial maps,
Category of objects]

True

class Algebras(category, *args)[source]#
class ParentMethods[source]#

Bases: object

group()[source]#

Return the underlying group of the group algebra.

EXAMPLES:

sage: GroupAlgebras(QQ).example(GL(3, GF(11))).group()                  # needs sage.groups sage.modules
General Linear Group of degree 3 over Finite Field of size 11
sage: SymmetricGroup(10).algebra(QQ).group()                            # needs sage.combinat sage.groups sage.modules
Symmetric group of order 10! as a permutation group

>>> from sage.all import *
>>> GroupAlgebras(QQ).example(GL(Integer(3), GF(Integer(11)))).group()                  # needs sage.groups sage.modules
General Linear Group of degree 3 over Finite Field of size 11
>>> SymmetricGroup(Integer(10)).algebra(QQ).group()                            # needs sage.combinat sage.groups sage.modules
Symmetric group of order 10! as a permutation group

class Finite(base_category)[source]#
class Algebras(category, *args)[source]#
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.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