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- Category_of_groups abstract "In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.There are two forgetful functors from Grp:M:Grp → MonU:Grp → SetWhere M has two adjoints:One right; I:Mon→GrpOne left; K:Mon→GrpHere I:Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K:Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid.The forgetful functor U:Grp → Set have a left adjoint given by the composite KF:Set→Mon→Grp where F is the free functor.The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms.The category Grp is both complete and co-complete. The category-theoretical product in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. The zero objects in Grp are the trivial groups (consisting of just an identity element).The category of abelian groups, Ab, is a full subcategory of Grp. Ab is an abelian category, but Grp is not. Indeed, Grp isn't even an additive category, because there is no natural way to define the "sum" of two group homomorphisms. (The set of morphisms from the symmetric group S3 of order three to itself, , has ten elements: an element z whose product on either side with every element of E is z (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always itself (the projections onto the three subgroups of order two), and six automorphisms. If Grp were an additive category, then this set E of ten elements would be a ring. In any ring, the zero element is singled out by the property that 0x=x0=0 for all x in the ring, and so z would have to be the zero of E. However, there are no two nonzero elements of E whose product is z, so this finite ring would have no zero divisors. A finite ring with no zero divisors is a field, but there is no field with ten elements because every finite field has for its order, the power of a prime.)Every morphism f : G → H in Grp has a category-theoretic kernel (given by the ordinary kernel of algebra ker f = {x in G | f(x) = e}), and also a category-theoretic cokernel (given by the factor group of H by the normal closure of f(H) in H). Unlike in abelian categories, it is not true that every monomorphism in Grp is the kernel of its cokernel.The notion of exact sequence is meaningful in Grp, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in Grp. The snake lemma however is not true in Grp.".
- Category_of_groups wikiPageID "800092".
- Category_of_groups wikiPageRevisionID "543489409".
- Category_of_groups hasPhotoCollection Category_of_groups.
- Category_of_groups subject Category:Category-theoretic_categories.
- Category_of_groups subject Category:Group_theory.
- Category_of_groups type Abstraction100002137.
- Category_of_groups type Category-theoreticCategories.
- Category_of_groups type Class107997703.
- Category_of_groups type Collection107951464.
- Category_of_groups type Group100031264.
- Category_of_groups comment "In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category.".
- Category_of_groups label "Categorie van groepen".
- Category_of_groups label "Category of groups".
- Category_of_groups label "Categoría de grupos".
- Category_of_groups label "Catégorie des groupes".
- Category_of_groups label "Категория групп".
- Category_of_groups sameAs Categoría_de_grupos.
- Category_of_groups sameAs Catégorie_des_groupes.
- Category_of_groups sameAs Categorie_van_groepen.
- Category_of_groups sameAs m.03d0nf.
- Category_of_groups sameAs Q3912012.
- Category_of_groups sameAs Q3912012.
- Category_of_groups sameAs Category_of_groups.
- Category_of_groups wasDerivedFrom Category_of_groups?oldid=543489409.
- Category_of_groups isPrimaryTopicOf Category_of_groups.