Editing Group homomorphism (section)

== Image and kernel ==
{{main article|Image (mathematics)|kernel (algebra)}}
We define the ''[[kernel (algebra)|kernel]] of h'' to be the set of elements in ''G'' which are mapped to the identity in ''H''
: <math> \operatorname{ker}(h) := \left\{u \in G\colon h(u) = e_{H}\right\}.</math>

and the ''[[Image (mathematics)|image]]  of h'' to be
: <math> \operatorname{im}(h) := h(G) \equiv \left\{h(u)\colon u \in G\right\}.</math>

The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The [[isomorphism theorem|first isomorphism theorem]] states that the image of a group homomorphism, ''h''(''G'') is isomorphic to the quotient group ''G''/ker ''h''.

The kernel of h is a [[normal subgroup]] of ''G''. Assume <math>u \in \operatorname{ker}(h)</math> and show <math>g^{-1} \circ u \circ g \in \operatorname{ker}(h)</math> for arbitrary <math>u, g</math>:
: <math>\begin{align}
  h\left(g^{-1} \circ u \circ g\right) &= h(g)^{-1} \cdot h(u) \cdot h(g) \\
                                       &= h(g)^{-1} \cdot e_H  \cdot h(g) \\
                                       &= h(g)^{-1} \cdot h(g) = e_H,
\end{align}</math>
The image of h is a [[subgroup]] of ''H''.

The homomorphism, ''h'', is a [[#monomorphism|''group monomorphism'']]; i.e., ''h'' is injective (one-to-one) if and only if {{nowrap|ker(''h'') {{=}} {''e''<sub>''G''</sub>}}}. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:
:<math>\begin{align}
                  &&                           h(g_1) &= h(g_2) \\
  \Leftrightarrow &&         h(g_1) \cdot h(g_2)^{-1} &= e_H \\
  \Leftrightarrow && h\left(g_1 \circ g_2^{-1}\right) &= e_H,\  \operatorname{ker}(h) = \{e_G\} \\
  \Rightarrow     &&               g_1 \circ g_2^{-1} &= e_G \\
  \Leftrightarrow &&                              g_1 &= g_2 
\end{align}</math>