I have to prove that $S_{4}$ is not isomorphic to $C_{2}\times A_{4}$ and I have no idea to do it. Some ideas?
I have tried a lot of methods but with no luck.
I have to prove that $S_{4}$ is not isomorphic to $C_{2}\times A_{4}$ and I have no idea to do it. Some ideas?
I have tried a lot of methods but with no luck.
$C_2\times A_4$ has an element of order 6. $S_4$ doesn't.
(On the other hand, $S_4$ has an element of order 4, which $C_2\times A_4$ doesn't).
Check that $S_4$ does not admit a surjective homomorphism onto $A_4$, because $S_4$ has no normal subgroup of order $2$.
If $S_4$ were isomorphic to $C_2\times A_4$ via $f\colon C_2\times A_4\to S_4$, then the image of $(1,0)$ would be an element of order $2$ that is central (commutes with everything) in $S_4$.
In $S_4$, there are two types of elements of order $2$: transpositions, and products of two disjoint transpositions.
If $i,j\in\{1,2,3,4\}$, $i\lt j$, does $(ij)$ commute with everything in $S_4$?
If $(ij)(k\ell)\in S_4$ is a product of two disjoint transpositions, does it commute with everything in $S_4$?