TITLE: Show $S_4$ is not isomorphic to $D_{24}$ by looking at their centers
QUESTION [6 upvotes]: Every proof seems to use an argument that looks at the orders and finds an element with a certain order in $S_4$ and no element in $D_{24}$ has that order...
Wouldn't it be much much easier to look at the centers? The center of $S_4$ is just $\{e\}$ but the center of $D_{24}$ has two elements in it... thus an isomorphism cannot exist.

REPLY [3 votes]: (Just so this question has an answer.)
That is of course a valid proof. Whether it would be easier is debatable. Showing the center of $D_{24}$ contains two elements isn't too bad as there is a standard generating set with simple relations. I don't know how easy it is to show that $S_4$ has trivial center. Furthermore, the definition of order of an element is usually presented earlier than the definition of the center of a group, so the proofs you've seen may be considered more elementary.