I don't really have any good thoughts on how to start this, so I've been looking at a possible contradiction proof or the contrapositive statement, which is all I'm asking about for now. Is the contrapositive statement: "Show that if F is not finite then $\displaystyle GL_n(F)$ is not finite"? The reason that I'm not sure is that it seems vastly easier to prove than the original statement, which is something I really didn't expect. I expected them to be about the same difficulty.Let F be a field. Show that if $\displaystyle GL_n(F)$ is finite then F is also finite.

Thanks!

-Dan