$2$-Sylow subgroups and $3$-Sylow subgroups of $A_4$

Question

I am working on finding all $2$- and $3$-Sylow subgroups of $A_{4}$. The answer I got is:

$2$-Sylow : $\langle(12)(34), (14)(23)\rangle$

$3$-Sylow : $\langle(123)\rangle, \langle(124)\rangle, \langle(134)\rangle, \langle(234)\rangle$

Is the answer right and exhaustive? Thanks.

Answer 1

Here's the full list. They're isomorphic to Klein's ViererGruppe:

• $\bigl\{(), (12), (34), (12)(34)\bigr\}$,

• $\bigl\{(), (13), (24), (13)(24)\bigr\}$,

• $\bigl\{(), (14), (23), (14)(23)\bigr\}$.

Answer 2

Hint: Remember third Sylow theorem, in particular

$$n_p | m, n_p \equiv 1 \mod p$$

and revise your answer for $p=2$