Editing Isomorphism theorems (section)

=== Discussion ===

The first isomorphism theorem can be expressed in [[category theory|category theoretical]] language by saying that the [[category of groups]] is (normal epi, mono)-factorizable; in other words, the [[normal morphism|normal epimorphisms]] and the [[monomorphism]]s form a [[factorization system]] for the [[category (mathematics)|category]].  This is captured in the [[commutative diagram]] in the margin, which shows the [[object (category theory)|objects]] and [[morphism]]s whose existence can be deduced from the morphism <math> f : G \rightarrow H</math>.  The diagram shows that every morphism in the category of groups has a [[Kernel (category theory)|kernel]] in the category theoretical sense; the arbitrary morphism ''f'' factors into <math>\iota \circ \pi</math>, where ''ι'' is a monomorphism and ''π'' is an epimorphism (in a [[conormal category]], all epimorphisms are normal).  This is represented in the diagram by an object <math>\ker f</math> and a monomorphism <math>\kappa: \ker f \rightarrow G</math> (kernels are always monomorphisms), which complete the [[short exact sequence]] running from the lower left to the upper right of the diagram.  The use of the [[exact sequence]] convention saves us from having to draw the [[zero morphism]]s from <math>\ker f</math> to <math>H</math> and <math>G / \ker f</math>.

If the sequence is right split (i.e., there is a morphism ''σ'' that maps <math>G / \operatorname{ker} f</math> to a {{pi}}-preimage of itself), then ''G'' is the [[semidirect product]] of the normal subgroup <math>\operatorname{im} \kappa</math> and the subgroup <math>\operatorname{im} \sigma</math>.  If it is left split (i.e., there exists some <math>\rho: G \rightarrow \operatorname{ker} f</math> such that <math>\rho \circ \kappa = \operatorname{id}_{\text{ker} f}</math>), then it must also be right split, and <math>\operatorname{im} \kappa \times \operatorname{im} \sigma</math> is a [[direct product]] decomposition of ''G''.  In general, the existence of a right split does not imply the existence of a left split; but in an [[abelian category]] (such as [[category of abelian groups|that of abelian groups]]), left splits and right splits are equivalent by the [[splitting lemma]], and a right split is sufficient to produce a [[Direct sum of groups|direct sum]] decomposition <math>\operatorname{im} \kappa \oplus \operatorname{im} \sigma</math>.  In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence <math>0 \rightarrow G / \operatorname{ker} f \rightarrow H \rightarrow \operatorname{coker} f \rightarrow 0</math>.

In the second isomorphism theorem, the product ''SN'' is the [[join and meet|join]] of ''S'' and ''N'' in the [[lattice of subgroups]] of ''G'', while the intersection ''S''&nbsp;∩&nbsp;''N'' is the [[join and meet|meet]].

The third isomorphism theorem is generalized by the [[nine lemma]] to [[abelian categories]] and more general maps between objects.