I have a problem counting the number of ways one can seat people around a circular table.

When ordering does not matter, $n$ people can sit around the table $n!$ ways, and then, because rotations are over counted and there are $n$ rotations of the table, we have that the number of ways is $(n-1)!$.

Now, when saying that if $n$ was an even number with half women and half men, how many ways to arrange them if no one can sit beside someone of the same sex?

I have read answers online, and they all say arrange the $(n/2)!$ women in a circle, divide by the number of rotations so you get $(n/2-1)!$ and then arrange the men so that you get $(n/2-1)!(n/2)!$. 

My question here is: Why do we not divide by $n$ as before, if we once again have all those extra rotations?