Prove that K is subgroup.

Question

Let K be the set of all permutations of $ S_4 $ type $ [2 ^ 2] $ and the identity permutation $\in K $. Prove that $ K $ is a subgroup $ S_4 $. I would like to prove that for$\pi, p \in S_4$ and type of $[2^2]$
$\pi p $ also is in K. But I can't deal with it.

Answer 1

Of the $24$ permutations in $S_4$ there are exactly $3$ permutations of the desired type. Write them explicitly, and then verify that if you multiply two of them you get the third.

The above shows that $K$ is closed under multiplication, to finish the argument that $K$ is a subgroup show that each of your $3$ non-trivial elements is its own inverse and thus $K$ is also closed under taking inverses.