How can I prove the following statement:
$S_n$ and $A_n \times \mathbb{Z}_2$ are not isomorphic for every $n>2$.
How can I prove the following statement:
$S_n$ and $A_n \times \mathbb{Z}_2$ are not isomorphic for every $n>2$.
A simple idea is to count the number of elements with order $2$ in the two groups. $g\in S_n$ has order $2$ iff it is product of disjoint transpositions, $h=(j,k)\in A_n\times\mathbb{Z}_2$ has order $2$ if $j$ is a product of an even number of disjoint transpositions, so, if $n\geq 4$, the elements of order $2$ in $S_n$ are striclty more than the elements of order $2$ in $A_n\times\mathbb{Z}_2$ and the two groups cannot be isomorphic. For $n=3$ we have that $A_3\times \mathbb{Z}_2$ is isomorphic to the abelian group $\mathbb{Z}_6$ while $S_3$ is not an abelian group.
Another way to see that $S_n$ and $A_n\times \mathbb{Z}/2$ are not isomorphic is that the latter contains an element of order $2$ which commutes with every other element. However, if $n>2$ the only element of $S_n$ that commutes with all other elements is the identity. (To see this, write down the cycle decomposition of your candidate permutation, and pick a transposition that either intersects two cycles or fits inside a cycle of length at least $3$.)
In other words, $A_n\times \mathbb{Z}/2$ has a nontrivial center, while the center of $S_n$ is trivial.
One way to do this is to show that $S_n$ does not have a normal subgroup of size 2 (since the direct product has such a normal subgroup).
The non-trivial element in such a subgroup would be a product of disjoint transpositions. But conjugating such a thing by any transposition which doesn't occur in the permutation but which does contain a digit that occurs in the permutation will change it, and we can ensure such a transposition exists because $n > 2$.
If you take $Q=\langle (1,2)\rangle\subset S_n$ then it can be a complement of $A_n$ in $S_n$. So $S_n$ is a semidirect product of $A_n$ by $Q$.