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

class Algebras(category, *args)#
class ParentMethods#

Bases: object

group()#

Return the underlying group of the group algebra.

EXAMPLES:

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

class Finite(base_category)#
class Algebras(category, *args)#
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