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Kernel of a group homomorphism, which is a Normal subgroup of the domain.

In fact normal subgroups can be characterized as kernels of homomorphisms

Image of a group homomorphism, which is a subgroup of the codomain

Group isomorphism

Homomorphism theorem

The Group automorphisms of a group form themselves a group, denoted as $Aut(G)$. It is a Subgroup of $S(G)$, the set of all bijections from G to itself (Symmetric group).

Inner automorphism. The set of inner automorphisms $Inn(G) \simeq G/Z(G)$ (isomorphic), where $Z(G)$ is the Center of a group