Group Isomorphism - Associativity - Change of Operator

Question

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.

What are the properties of a group? What is a group isomorphism?
Note that commutativity is not a defining property of a group, rather it is an additional property a group can have.
Please note the obvious misprint, you surely mean $a *' b = b * a$.
@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 I made no comment about the relevance of commutativity. What I said was it is not a basic property of a group.

Ben · Accepted Answer · 2013-01-26 21:32:00Z

As this is a homework problem, here are some hints.

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.

1 Answer