I am trying to list all the cyclic subgroups of $S_4$, and also list two examples of proper non-cyclic subgroups of $S_4$.

Here's the general idea of what I have for the list: $S_4$, $A_4$, $\langle e\rangle$,

$\langle(1 2)\rangle, \langle(2 3)\rangle$, and so on...

$\langle(134)\rangle, \langle(234)\rangle$, and so on...

$\langle(12)(34)\rangle, \langle(13)(24)\rangle$, and so on...

$\langle(234)(23)\rangle, \langle(123)(12)\rangle$ and so on...

You get the idea. There are thirty in total by my count.

Is this $i.$ proper notation for a subgroup of $S_4$ and $ii.$ how would I find a proper non-cyclic subgroup?

  • 2
    $\begingroup$ $S_4$ has twenty-four elements. How can it have more than twenty-four cyclic subgroups? $\endgroup$ – Angina Seng Nov 9 '19 at 19:54
  • 2
    $\begingroup$ Use $\langle X\rangle$ for $\langle X\rangle$. $\endgroup$ – Shaun Nov 9 '19 at 19:58
  • $\begingroup$ @LordSharktheUnknown, you are correct, there are 30 total subgroups but not all of them are cyclic, that is my mistake. There are only 17 cyclic subgroups, $\endgroup$ – o's1234 Nov 9 '19 at 20:26