Let $S$ be a set with an element $O$ and a composition law $*$ satisfying the following two conditions:
- $P*Q=Q*P$ for all $P,Q\in S$
- $P*(P*Q)=Q$ for all $P,Q\in S$.
Define an operation $+$ on $S$ by $P+Q=O*(P*Q)$, and assume that $+$ is associative, which is equivalent to the condition that
$P*(O*(R*Q))=R*(O*(Q*P))$ for all $P,Q,R\in S$.
This makes (S,+) into a group. Now define a new operation $+'$ on $S$ which is also associative, but with $O'$ in place of $O$. Prove that the function $F:(S,+)\rightarrow (S,+')$ defined by $P\mapsto O*(O'*P)$ is a group isomorphism.
Whew! That was a lot to write down. Ok so bijectivity is easy, but I'm unable to prove the homomorphism part. I have:
Now the only non trivial move I can see to make at this point is to use the associativity condition to write this as:
From here I can't find anything more to do despite a lot of trying, and I certainly can't make it look even close to: