> That assumes associativity, but that's a nitpick, not a real objection.

I don't think that is a valid nitpick. My earlier comments assume associativity because a group operation is associative by definition. If we do not allow associativity, then the algebraic structure we are working with is no longer a group at all. It would just be a loop (which is a quasigroup which in turn is magma).

> Thanks for taking the time for the thoughtful, and non-snarky, response. Sorry if I was abrupt before.

No worries at all. I'm glad to have a place on the Internet where I can talk about these things now and then. Thank you for engaging in the discussion.

You are again right. I misrecalled a group as a loop.

Thank you again. It's been too long since I've had to use this knowledge and am happy to have the opportunity to (try to) use it.