Additive groups

class sage.categories.additive_groups.AdditiveGroups(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

The category of additive groups.

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
sage: from sage.categories.additive_monoids import AdditiveMonoids
sage: AdditiveGroups()
Category of additive groups
sage: AdditiveGroups().super_categories()
[Category of additive inverse additive unital additive magmas,
 Category of additive monoids]
sage: AdditiveGroups().all_super_categories()
[Category of additive groups,
 Category of additive inverse additive unital additive magmas,
 Category of additive monoids,
 Category of additive unital additive magmas,
 Category of additive semigroups,
 Category of additive magmas,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]

sage: AdditiveGroups().axioms()
frozenset({'AdditiveAssociative', 'AdditiveInverse', 'AdditiveUnital'})
sage: AdditiveGroups() is AdditiveMonoids().AdditiveInverse()
True

AdditiveCommutative

alias of sage.categories.commutative_additive_groups.CommutativeAdditiveGroups

class Algebras(category, *args)

Bases: sage.categories.algebra_functor.AlgebrasCategory

class ParentMethods

Bases: object

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: SymmetricGroup(10).algebra(QQ).group()
Symmetric group of order 10! as a permutation group

class Finite(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

class Algebras(category, *args)

Bases: sage.categories.algebra_functor.AlgebrasCategory

class ParentMethods

Bases: object

extra_super_categories()

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