Group (mathematics)
Quotient groups

Suppose that N {\displaystyle N} is a normal subgroup of a group G {\displaystyle G}, and G / N = { g N ∣ g ∈ G } {\displaystyle G/N=\{gN\mid g\in G\}} denotes its set of cosets.Then there is a unique group law on G / N {\displaystyle G/N} for which the map G → G / N {\displaystyle G\to G/N} sending each element g {\displaystyle g} to g N {\displaystyle gN} is a homomorphism.Explicitly, the product of two cosets g N {\displaystyle gN} and h N {\displaystyle hN} is (g h) N {\displaystyle (gh)N}, the coset e N = N {\displaystyle eN=N} serves as the identity of G / N {\displaystyle G/N}, and the inverse of g N {\displaystyle gN} in the quotient group is (g N) − 1 = (g − 1) N {\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}.The group G / N {\displaystyle G/N}, read as " G {\displaystyle G} modulo N {\displaystyle N} ", is called a quotient group or factor group.The quotient group can alternatively be characterized by a universal property.

The elements of the quotient group D 4 / R {\displaystyle \mathrm {D} _{4}/R} are R {\displaystyle R} and U = f v R {\displaystyle U=f_{\mathrm {v} }R}. The group operation on the quotient is shown in the table. For example, U ⋅ U = f v R ⋅ f v R = (f v ⋅ f v) R = R {\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}. Both the subgroup R = { i d, r 1, r 2, r 3 } {\displaystyle R=\{\mathrm {id},r_{1},r_{2},r_{3}\}} and the quotient D 4 / R {\displaystyle \mathrm {D} _{4}/R} are abelian, but D 4 {\displaystyle \mathrm {D} _{4}} is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the semidirect product construction; D 4 {\displaystyle \mathrm {D} _{4}} is an example.

The first isomorphism theorem implies that any surjective homomorphism ϕ: G → H {\displaystyle \phi:G\to H} factors canonically as a quotient homomorphism followed by an isomorphism: G → G / ker ϕ → ∼ H {\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}.Surjective homomorphisms are the epimorphisms in the category of groups.
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