Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$

Question

Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$

I have no idea how to show this. I'm studying for a test, so I am less interested in solutions and hints than I am strategy. What should my thought process be as I try to solve this problem? What kind of thing do I look for?

Answer 1

We want to consider the properties that are invariant under isomorphism and see that there is one that the two groups do not share.

For example, $D_4\times\mathbb{Z}_3$ has an element of order $3$ in its center. The elements of order $3$ in $S_4$ are the $3$-cycles, none of which are in the center.

For another example, $D_4\times \mathbb{Z}_3$ has an element of order $6$. $S_4$ has no elements of order $6$.

For yet another example, $D_4\times\mathbb{Z}_3$ has $5$ elements of order $2$. $S_4$ has $(12),(13),(14),(23),(24),(34),\ldots$ which is already too many.

The list goes on.