I am trying to solve a problem of Hamermesh's Group Theory and its Applications to Physical Problems.
Problem: The cyclic permutations on four symbols form a subgroup $H$ of $S_4$. Resolve $S_4$ into left cosets with respect to $H$.
What are the cyclic permutation that form the subgroup $H$? I understand that it would be $(1234)$,$(1243)$,$(1342)$,$(1324)$,$(1432)$ and $(1423)$. But these don't form a subgroup as far as I understand. So I'm pretty sure that misunderstand what are the members of the subgroup $H$.
Can someone help me?