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Fraleigh defines an isomorphism as:

Let $\langle S, * \rangle$ and $\langle S, * \rangle$ be binary algebraic structures. An isomorphism of $S$ with $S'$ is a one-to-one function $\phi$ mapping $S$ onto $S'$ such that $$\phi(x*y) = \phi(x) *' \phi(y) \text{ for all x, y} \in S$$

The author then describes a recipe for showing that two structures are isomorphic. His third step (after "define $\phi$" and "show $\phi$ is 1-1") is to show that $\phi$ is onto $S'$.

Why is this necessary? It makes sense to me that two structures to need to have the same cardinality for isomorphism, but that's not rigorous. From where in the definition does surjection arise?

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    $\begingroup$ Onto-ness arises in the definition when the author says that $\phi$ maps $S$ onto $S'$. $\endgroup$
    – user61527
    Dec 23 '13 at 4:10
  • $\begingroup$ Wow! I am BLIND. Thank you. $\endgroup$
    – xzackli
    Dec 23 '13 at 4:13
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T. Bongers has answered this question in the comments.

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  • $\begingroup$ Belated question: do I accept this as an answer? If an SE question is resolved, ought there be an acceptance? $\endgroup$
    – xzackli
    Dec 31 '13 at 19:38
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    $\begingroup$ @ZackLi It would be helpful for you to accept the answer, although the most important thing for the site is that at least one answer was upvoted. (Questions with no upvoted answers appear in the "unanswered" questions list, and I think they are ocassionally bumped to the front page.) Because I have marked this answer as "community wiki", I will not receive points for either the upvote or the acceptance. $\endgroup$
    – Jim Belk
    Dec 31 '13 at 21:39

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