Proof that $o(\sigma)=3$ $\Rightarrow$ $\sigma \in$ Conjugacy Class of $(1 2 4)$ How can I show that $o(\sigma)=3$ $\Rightarrow$ $\sigma \in$ C($(1 2 4))$ in $S_4$? All I have to go with is that $\sigma ^3 = I$. I considered the use of Cayley's theorem, which would mean that a group like $\{\sigma, \sigma ^2, \sigma ^3 \}$ would be isomorphic to a subgroup of the symmetric group. Could I then go on to claim that somehow the conjugacy class is isomorphic to  $S_4$?

Since $\sigma$ is of order $3$ in $S_4$, therefore $\sigma$ must be of the form $( a\, b \, c)(d)$ (I will now drop the $1-$cycle). For $\sigma$ to be in $C(1\,2\,4)$ we need the existence of a $\beta \in S_4$ such that 
$$\beta (1\, 2\, \,4) \beta^{-1}=\sigma=(a\,b\,c)$$
The LHS is $\big(\beta(1) \, \beta(2) \, \beta(4)\big)$. So your permutation $\beta$ should be such that $1 \stackrel{\beta}\to a$, $2 \stackrel{\beta}\to b $ and $4 \stackrel{\beta}\to c$ and clearly such a $\beta \in S_4$. Thus $\sigma$ is a conjugate of $(1\, 2 \, 4)$.
