this is a quite popular question is ring theory i guess. there's a solution which uses the structure theory of rings:
the answer is you don't need to assume is finite. it's clear that for there exist rings with now let be a ring and suppose then in because
otherwise would be a subgroup of which is impossible because is odd. now look at the group algebra by Maschke's theorem is semisimple and thus by Wedderburn-Artin
theorem where each is a division ring containing (here means the ring of matrices with entries from ) since is finite, each is
finite. but we know that a finite division ring is a field. so each is a finite field extension of i.e. for some
also if for some then which is odd only for thus but then and
hence: which is clearly impossible. Q.E.D.
Remark: the above solution uses some powerful theorems in ring theory and so you should expect to get a lot more than just answering your question out of it. so ... see if you can do that!
Briliant!
Over in the AoPS/MathLinks forum at the moment, some people are trying to determine if there exists a finite ring with unity such that The latest development appears to be that if such a ring exists, it must have characteristic 4. It looks exciting – I wish I knew more ring theory so I could join in and not be left out of the fun.