Let A,B be algebraic structures and let h A-->B be a bijective homomorphism.

Is h an isomorphism? In topology, we have continuous bijections that are not homeomorphisms,

(similar in Functional Analysis )so I wondered if the "same" was possible in Algebra. I assume if there is a counterexample, it requires an infinite set in the construction, or some result in order theory, or some issue with torsion .

Thanks,

WWGD: "What Would Gauss Do?".