Finite groups¶
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class
sage.categories.finite_groups.
FiniteGroups
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom
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
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class
Algebras
(category, *args)¶ Bases:
sage.categories.algebra_functor.AlgebrasCategory
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class
ParentMethods
¶ Bases:
object
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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
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class
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class
ElementMethods
¶ Bases:
object
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class
ParentMethods
¶ Bases:
object
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cardinality
()¶ Returns the cardinality of
self
, as perEnumeratedSets.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 oforder()
.EXAMPLES:
We need to use a finite group which uses this default implementation of cardinality:
sage: G = groups.misc.SemimonomialTransformation(GF(5), 3); G Semimonomial transformation group over Finite Field of size 5 of degree 3 sage: G.cardinality.__module__ 'sage.categories.finite_groups' sage: G.cardinality() 384
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cayley_graph_disabled
(connecting_set=None)¶ AUTHORS:
Bobby Moretti (2007-08-10)
Robert Miller (2008-05-01): editing
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conjugacy_classes
()¶ 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 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
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conjugacy_classes_representatives
()¶ Return a list of the conjugacy classes representatives of the group.
EXAMPLES:
sage: G = SymmetricGroup(3) sage: G.conjugacy_classes_representatives() [(), (1,2), (1,2,3)]
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monoid_generators
()¶ 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 ((2,3,4), (1,2,3))
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semigroup_generators
()¶ Returns 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 ((2,3,4), (1,2,3))
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some_elements
()¶ Return some elements of
self
.EXAMPLES:
sage: A = AlternatingGroup(4) sage: A.some_elements() Family ((2,3,4), (1,2,3))
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example
()¶ 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
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class