# Group Isomorphism - Associativity - Change of Operator

Sorry about the title - I wasn't sure about how to be more specific.

This is a homework problem, where I've only been able to write down (ii).

Let (S,*) be a binary system. Define the opposite operation *' to * by a*'b=b*a.

• (i) Prove that if * is associative, then so is *'.

• (ii) Give an example where (S, *) and (S, *') are not isomorphic. <--- Done.

• (iii) Prove that if (S,*) is a group, then (S, *') and (S, *) are isomorphic.

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What are the properties of a group? What is a group isomorphism? – Erik G. Jan 26 at 20:37
Note that commutativity is not a defining property of a group, rather it is an additional property a group can have. – Erik G. Jan 26 at 20:38
Please note the obvious misprint, you surely mean $a *' b = b * a$. – Andreas Caranti Jan 26 at 20:43
@ErikG.: Part (iii) is true whether $(S,*)$ is abelian or not (but if it's not abelian, then the identity map is not an isomorphism). – Brad Jan 26 at 20:58
@Brad I made no comment about the relevance of commutativity. What I said was it is not a basic property of a group. – Erik G. Jan 26 at 21:00
For (i), evaluate $(a*'b)*'c$ and $a*'(b*'c)$, keeping in mind that $*$ is associative, and show that they are equal. Then $*'$ is associative.
For (iii), try verifying that $f\colon (S,*)\to (S,*'):x\mapsto x^{-1}$ is a group isomorphism.