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As one goes deeper in abstract algebra, for example in topics like algebraic topology, there are often arguments when the authors just write things like

So we have Z + Z / (2,-2)Z is isomorphic to Z + (Z/2Z)

or

Z cannot be isomorphic to Z + 2/Z

where they don't describe the map between those two objects or don't give an argument. So they have to "see directly" that this is possible or not.

My question is now how one can see faster such relations. I think the second one is pretty clear, but the first one is not intuitive for me.

Thanks for your help, have a nice day!

Shaun
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mkfrnk
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  • Are you aware of the classification of finitely generated abelian groups? The ones you mention above are probably all implicitly mapping to the standard presentation, from which one can readily deduce if two groups are isomorphic (if they are both finitely generated abelian). – Dan Rust Aug 02 '22 at 09:30
  • See also this post, and others, for quotient groups of $\Bbb Z\oplus \Bbb Z$, and how to find the isomorphisms. – Dietrich Burde Aug 02 '22 at 10:07
  • The method is precisely the same you adopt when you are asked for the result of $6\times 9$, or $7\times8$. It is called mileage! – Ruy Aug 02 '22 at 19:47

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