Editing Kernel (algebra) (section)

=== Group homomorphisms ===
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Let ''G'' and ''H'' be [[group (mathematics)|group]]s and let ''f'' be a [[group homomorphism]] from ''G'' to ''H''. If ''e''<sub>''H''</sub> is the [[identity element]] of ''H'', then the ''kernel'' of ''f'' is the preimage of the singleton set {''e''<sub>''H''</sub>}; that is, the subset of ''G'' consisting of all those elements of ''G'' that are mapped by ''f'' to the element ''e''<sub>''H''</sub>.<ref name="Dummit Group Kernel Definition"/><ref name="Hungerford Kernel">{{harvnb|Hungerford|2014|p=263}}</ref>

The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation).<ref name="Dummit Group Kernel Definition"/> In symbols:
: <math> \ker f = \{g \in G : f(g) = e_{H}\} .</math>

Since a group homomorphism preserves identity elements, the identity element ''e''<sub>''G''</sub> of ''G'' must belong to the kernel.<ref name="Dummit Group Kernel Definition"/> The homomorphism ''f'' is injective if and only if its kernel is only the singleton set {''e''<sub>''G''</sub>}.<ref>{{harvnb|Hungerford|2014|p=264}}</ref>

{{nowrap|ker ''f''}} is a [[subgroup]] of ''G'' and further it is a [[normal subgroup]]. Thus, there is a corresponding [[quotient group]] {{nowrap|''G'' / (ker ''f'')}}. This is isomorphic to ''f''(''G''), the image of ''G'' under ''f'' (which is a subgroup of ''H'' also), by the [[isomorphism theorems|first isomorphism theorem]] for groups.<ref name="Dummit Group FIT"/>