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The above excerpt is from Herstein's book "Topics in Algebra". It confuses me by two reasons: Firstly, in the first definition the mapping is into but in the second the mapping is onto. Why the author uses into and onto?

Could anyone clarify this distinction in definitions, please?

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    $\begingroup$ What is $\overline{G}$? The image of $\phi$? $\endgroup$ – Rellek Dec 23 '17 at 16:11
  • $\begingroup$ @Rellek, I guess that $\bar{G}$ is group and $\phi$ is the mapping betweeb $G$ and $\bar{G}$ $\endgroup$ – ZFR Dec 23 '17 at 16:25
  • $\begingroup$ @Rellek "A mapping $\phi$ from a group $G$ into a group $\bar{G}$ is said to be a homomorphism if for all $a, b \in G,\ \phi( ab) = \phi(a)\phi (b)$", so according to Herstein it is just another group. $\endgroup$ – The Phenotype Dec 23 '17 at 16:26
  • $\begingroup$ Perhaps it is simply a typo? Seems unlikely for such a fundamental definition, but maybe Herstein meant to write onto instead of into. $\endgroup$ – Rellek Dec 23 '17 at 16:28
  • $\begingroup$ @Rellek : From a page further back, it's just some codomain group: "Defn.: A mapping $\phi$ from a group $G$ into a group $\bar{G}$ is said to be a homomorphism if for all $a,b \in G$, $\phi(ab) = \phi(a)\phi(b).$" $\endgroup$ – Eric Towers Dec 23 '17 at 16:34
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In Herstein's terminology (which is rarely used nowadays) an isomorphism is just an injective homomomorphism.

On the other hand, two groups $G$ and $G'$ are said to be isomorphic if there exists a surjective (onto) isomorphism $\phi\colon G\to G'$.

Herstein uses the preposition “into” to generically introduce the codomain, so a map $f\colon X\to Y$ is from $X$ into $Y$. The preposition “onto” is used when the map is surjective.

The terminology used by Herstein is quite old-fashioned and now it's commonly preferred to say “injective homomorphism” or “monomorphism” instead of “isomorphism”. Note that, in Herstein's terminology, the existence of an isomorphism $\phi\colon G\to G'$ doesn't imply that $G$ and $G'$ are isomorphic. Only the existence of an “onto isomorphism“ does.

Herstein's book was first published in 1975; at the time terminology was still unsettled, but my algebra teachers always used “isomorphism” for “bijective homomorphism”. It's always best to check the book for definitions and usage and keep a “dictionary” for translations.

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  • $\begingroup$ I cannot the understand the meaning of third paragraph of your post. Could you clarify it please? $\endgroup$ – ZFR Dec 23 '17 at 17:18
  • $\begingroup$ @RFZ It's just an explanation of the usage: the symbol $f\colon X\to Y$ is read “$f$ is a map (or homomorphism) of/from $X$ into $Y$” and has no other implications on $f$, unless other conditions are stated. When the words “$f$ is a map of/from $X$ onto $Y$” are used, then it is implied that $f$ is surjective. $\endgroup$ – egreg Dec 23 '17 at 17:21
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    $\begingroup$ @RFZ Yes, it's ambiguous; that's why modern usage is different. Note that the second definition doesn't deal with the term “isomorphism”, but with the relation “being isomorphic”. $\endgroup$ – egreg Dec 23 '17 at 18:14
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    $\begingroup$ @RFZ I don't understand. While ambiguous for who's accustomed with modern usage, Herstein's terminology is used consistently across the book. $\endgroup$ – egreg Dec 23 '17 at 18:20
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    $\begingroup$ @RFZ Unfortunately, Herstein uses the word “isomorphism“ in the sense specified in the first definition; if you change it, the book would contain contradictions. It's what it is: when you find “isomorphism” in the book, pretend it's written “monomorphism“. $\endgroup$ – egreg Dec 23 '17 at 18:24

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