Generating a subgroup

Question

While getting ready for my exams I am trying to solve this question

"Find a Sylow $2$-subgroup in $S_4$ which contains the element $(1, 3, 2, 4)$."

I started by permutation of the element required on itself till i got the identity element. thus i have $4$ elements but i need $4$ more. how do i know which ones they are?

(i.e. what is the process of getting the full sub group?)

Thanks in advance!

Answer 1

Set $H=<(1,2,3,4)>$ then $|H|=4$ and notice that $S_4$ has a normal subgroup of $N$ order $4$ which is isomorphic to Klein four group. $HN$ is a group since $N$ is normal and since $|H\cap N|=2$ then $|HN|=4.4/2=8$. Thus, $HN$ is the desired subgroup.

Note: $N=\{(1),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$