> 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 becau

> 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.