Definition of isomorphism from Herstein's book 
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?
 A: 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.
