Google Groups is one place where, IMHO, Google pushes its hegemony too far, making it difficult to use. I wanted to subscribe to puppet-users with my Mozilla address, but since I have a Google account, Groups assumes I want to subscribe with that address. No!
I found the fix with a bit of Googling (some irony there). It involves editing a URL:
where you’d substitute the name of the group you want for puppet-users and add your email at the end.