# The definition of a normal subgroup

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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Equivalent definitions usually require proof of equivalence. – EuYu Oct 17 '12 at 23:33
This is precisely the first isomorphism theorem. If we define a normal subgroup $N$ to be a subgroup of $G$ such that $gNg^{-1}\in N$ for all $g\in G$ then the first isomorphism theorem states that every normal subgroup is the kernel of some homomorphism and the kernel of every homomorphism is a normal subgroup. Thus, they are equivalent. – user1729 Oct 18 '12 at 11:08

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 $G$.

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You write "its normal subgroup" -- this would mean a group can only ever have one normal subgroup. If you don't mean it this way you could write something like "$N$ a normal subgroup of $G$". – Rudy the Reindeer Oct 20 '12 at 15:16
@MattN. Thanks for the English grammar. – Makoto Kato Oct 20 '12 at 20:26