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- Quotient_group abstract "In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by aggregating similar elements of a larger group using an equivalence relation. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity.In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G / N, where G is the original group and N is the normal subgroup. (This is pronounced "G mod N," where "mod" is short for modulo.)Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism φ: G → H is isomorphic to G / ker(φ) where ker(φ) denotes the kernel of φ.The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects. For other examples of quotient objects, see quotient ring, quotient space (linear algebra), quotient space (topology), and quotient set.".
- Quotient_group wikiPageID "11526".
- Quotient_group wikiPageRevisionID "600492423".
- Quotient_group hasPhotoCollection Quotient_group.
- Quotient_group subject Category:Group_theory.
- Quotient_group comment "In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by aggregating similar elements of a larger group using an equivalence relation.".
- Quotient_group label "Factorgroep".
- Quotient_group label "Faktorgruppe".
- Quotient_group label "Groupe quotient".
- Quotient_group label "Grupa ilorazowa".
- Quotient_group label "Grupo cociente".
- Quotient_group label "Grupo quociente".
- Quotient_group label "Gruppo quoziente".
- Quotient_group label "Quotient group".
- Quotient_group label "Факторгруппа".
- Quotient_group label "زمرة خارج القسمة".
- Quotient_group label "商群".
- Quotient_group sameAs Faktorová_grupa.
- Quotient_group sameAs Faktorgruppe.
- Quotient_group sameAs Grupo_cociente.
- Quotient_group sameAs Groupe_quotient.
- Quotient_group sameAs Gruppo_quoziente.
- Quotient_group sameAs 몫군.
- Quotient_group sameAs Factorgroep.
- Quotient_group sameAs Grupa_ilorazowa.
- Quotient_group sameAs Grupo_quociente.
- Quotient_group sameAs m.031yl.
- Quotient_group sameAs Q1138961.
- Quotient_group sameAs Q1138961.
- Quotient_group wasDerivedFrom Quotient_group?oldid=600492423.
- Quotient_group isPrimaryTopicOf Quotient_group.