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\[
\operatorname{ker} f=\{g\in G:f(g)=e_{H}\}{\mbox{.}}
\]
The homomorphism \(f\) is injective if and only if its kernel is only the singleton set \(e_G\), because otherwise \(\exists a,b\in G\) with \(a\neq b\) such that \(f(a)=f(b)\).