Let $H = \{\text{id}, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) \} \subset S_4$, and let $ K = \{\sigma \in S_4 \mid \sigma(4) = 4\}.$
(a) How to show that $H$ is a subgroup of $S_4$, and is $H$ is a normal subgroup? What other group is $H$ isomorphic to? Same for $K.$
(b) Does every coset of $H$ contain exactly one element of $K.$ Also, can every element of $S_4$ be written uniquely as the product of an element of $H$ and an element of $K$?
(c) What can be said about the quotient group $S_4/H$? Is $S_4$ isomorphic to the direct product H × K?
For part a, to show that H is a subgroup of S4, I will take each element in S4 and conjugate it by what? i.e., to be normal NH = HN, or g-1ng = n
for part b, I am not sure what is being asked of