Showing that Klein Four Group is a normal subgroup of $S_4$

Question

It is given that $K_4=\{i, (1$ $2)(3$ $4), (1$ $3)(2$ $4), (1$ $4)(2$ $3)\}$. The question asks me to show that, for $h \in S_4$ and $f \in K_4$,
$$h^{-1}fh\in K_4,$$
using the order of the permutations to deduce the possible cycle-shapes of $h^{-1}fh$. I'm new to group theory so terms like isomorphic are foreign to me.

Answer 1

Hint: Show that the cycle structure of a permutation is preserved under conjugation.