Corollary 13.18

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A group homomorphism $\phi: G \rightarrow G'$ is a one-to-one map if and only if Ker$(\phi) = \{e\}$.

If Ker$(\phi) = \{e\}$, then for every $a \in G$, the elements mapped into $\phi(a)$ are precisely the elements of the left coset $a\{e\} = \{a\}$, which shows that $\phi$ is one to one.

Conversely, suppose $\phi$ is one to one. Now by Theorem 13.12, we know that $\phi(e) = e'$, the identity element of $G'$. Since $\phi$ is one to one, we see that $e$ is the only element mapped into $e'$ by $\phi$, so Ker$(\phi) = \{e\}$.

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