There are a whole bunch of equivalent definitions of a normal subgroup. Is it logically sound (with respect to the usual definitions) to include as a definition of normal, "$N$ is the kernel of some homomorphism $f$"?
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It's logically sound. Let $G$ be group, $N$ a normal subgroup of $G$. $N$ is the kernel of the canonical homomorphism $G \rightarrow G/N$. Conversely let $f\colon G \rightarrow G'$ be a group homomorphism. $Ker(f)$ is a normal subgroup of $G$.