What is the automorphism group of a complete bipartite graph isomorphic to?

Question

I couldn't quite figure out what the automorphism group of a complete bipartite graph $\Gamma (K_{r,s})$ is isomorphic to. I did some back-of-the-envelope calculations and found that $\Gamma (K_{r,s})$ has an order of $(s-1)!$ when $r = 1$ and $s \geq 2$, and has an order of $2(r!s!)$ when $ r, s \geq 2$ (I don't have a proof of that. I might be wrong.) but I'm not aware of any groups that might be isomorphic to $\Gamma (K_{r,s})$ as my knowledge of group theory is rather elementary.

Answer 1

Your computations of the number of automorphisms aren't quite right in general, though they are close.

Hints: Clearly, the $r$ nodes on one side are all equivalent to each other, and can be freely permuted; what is the symmetry group of $r$ identical objects? Given that similarly the $s$ nodes on the other are also all equivalent, we could consider an arbitrary permutation of those as well; how do we combine these two symmetry groups together to get a larger one? And finally, are there any special values of $r,s$ for which there is an extra symmetry?