def

$b \mapsto aba^{-1}$

$Inn(G)$ is the set if inner automorphisms. It is a Normal subgroup of the set of Group automorphisms (proof)

If G is commutative, then the inner automorphisms are just the identity. Inn(G) = {id_G}

$Inn(G) \simeq G/Z(G)$ (isomorphic), where $Z(G)$ is the Center of a group