$V_4\triangleleft S_4$

Question

Let $V_4:=\{(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3),\iota\} \leq S_4$.

It is possible to show $V_4\triangleleft S_4$ by considering conjugation.

However, after long thought on the matter, I don't see how one verifies that $$ v\in V_4,\sigma\in S_4\implies\sigma v\sigma^{-1}\in V_4 $$ without trying all possibilities. Of course this is trivial if $a=\iota$, and I suspect there must be an easy general argument.

Answer 1

Hint: In $S_n$, two permutations have the same cycle type if and only if they are conjugate.

Answer 2

Hint: if $v=(a,b,...,c)..(d,e,...,f),u \in S_n$ then $uvu^{-1}= (u(1),u(2),........,u(c))..(u(d),u(e),..,u(f))$. Use this fact to prove $V_4 \triangleleft S_4$.

Answer 3

$V_4$ consists of all the permutations with cycle length $(2,2)$ and the neutral element.