Let G be a group and suppose $g \in G$.
$\varphi:G\rightarrow Aut\left ( G \right )$
$g \mapsto i_{g}$
is a Homomorphism with image $Inn\left ( G \right )$ where $Inn\left ( G \right )=\left \{ i_{g}\mid g \in G \right \}$
The Kernel $Z\left ( G \right )=\left \{ x \in G \mid xg=gx, \forall x \in G \right \}$
Recall:
Let G be a group and suppose $g, h \in G.$
The conjugation by g, $i_{g}$ is defined as
$i_{g}:G\rightarrow G$
$h\mapsto g^{-1}hg$
Could someone explain why $ker(\varphi)$ is the centre of a group G?
Thanks in advance