You are perfectly correct, I was wrong. Upon closer reflection it appears impossible to have a nontrivial set (meaning other than {0}) that forms a group both under addition and under multiplication because the identity element of the former does not have an inverse in the latter. However, take this with a grain of salt, because I haven't had time to prove this to my own satisfaction yet (and I am loathe to go googling for answers in my own field of expertise).

My abstract algebra professor would be horrified by my misstatement, and I shall go to bring flowers as a sign of atonement and rememberance to his grave at the earliest available opportunity.