I graduated nearly two and half years ago and I didn’t keep in touch with my math instructors, so I thought maybe some of you fine folk could help me out. I’m trying to keep my algebra skills sharp but I’m having a bit of difficulty reconciling these two theorems.

Now it seems to me that the lemma is saying that you cannot take the product of an odd number of 2-cycles, which would not only be a contradiction of the theorem, but is demonstrably false: (abcd) = (ad)(ac)(ab) is a product of an odd number of 2-cycles. The proofs of both the lemma and the theorem are in my textbook, and I can follow them both just fine (or at least I think I can :smack:). So what am I missing?