Definition talk:Isomorphism (Abstract Algebra)/Group Isomorphism

Best definition?
I don't feel that the definition by bijectivity is the best definition. A better one, IMO, is that there are homomorphisms both ways, $\phi: G \to H$ and $\psi: H \to G$, which are inverse to each other. That way the definition states explicitly that the structure is preserved in both directions, so it is the same up to relabeling. It is then easily provable that the functions are bijections. I'm writing this from a broader perspective than group theory's, such as that of possibly multiple-valued operations, as in hyperfields, or structures like a topology or a graph, where I think the definition as a homomorphism that is a bijection is too weak to capture the meaning of isomorphism. I propose both definitions should be given. I'm (rightly, of course) reluctant to change this page without discussion. Zaslav (talk) 18:09, 20 April 2018 (EDT)


 * Feel free to add a second definition. We have on  many instances of concepts which have multiple definitions. What we do is post two definitions, each in separate sections (implemented as transcluded pages) and link them with an equivalence proof in a more-or-less standardised format.


 * If you can find a source work which explicitly provides the definition in the form you state it, then it would be good to document it here.


 * What I will also do as and when the mood takes me is to go through the listed sources and see whether they actually take the approach you suggest (I may have glossed over the difference when this page was originally posted). --prime mover (talk) 18:34, 20 April 2018 (EDT)