I want to show that ker f is a normal subgroup of some group $X$ Suppose I have two groups, call them $X$ and $Y$, and I let $f : X \longrightarrow Y$ be a group homomorphism. I want to prove that the ker $f$ is a normal subgroup of $X$. Here is my attempt. Let me know how my proof looks and if I am missing details:
We want to show that $aca^{-1} \in$ Ker $f$, $\forall a \in A$ and $\forall c \in$ Ker $f$.
$f(aca^{-1}) = f(a)f(c)f(a^{-1})$
$= f(a)f(c)f(a)^{-1}$
$= f(a)If(a)^{-1}$ because $c \in$ Ker $f$
$= f(a)f(a)^{-1}$
$= I$
Thus, ker $f$ is a normal subgroup of $X$.

It looks good, provided that you already know that the preimage of a subgroup is a subgroup. And it can easily be generalized to The preimage of a normal subgroup of $Y$ is a normal subgroup of $X$.
