This is a pet peeve of mine. Both my textbook and my instructor, often say $G$ is $G'$, when really what they mean is $G$ is isomorphic to $G'$. Why is this an accepted convention?
Why is it okay to say two groups are the same, when they're really isomorphic?
11
$\begingroup$
abstract-algebra
soft-question
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4Usually people do this when there is a canonical choice of isomorphism. – 2012-09-28
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4It depends on context. It is very important to be able to be both pedantic and flexible about this depending on the situation. – 2012-09-28
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0It is accepted convention because, in that context, the two objects are indistinguishable. (There *may* be things that distinguish them, but that will have to come from outside the context.) – 2012-09-28
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1Often things are only *defined* up to isomorphism, for example anything defined using category theory. – 2012-09-28
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3This seems very strange to me. Under what circumstances would you consider it to be correct to say that a group "is" $\Bbb Z_6$? Maybe you want to say that only $\{0,1,\ldots 5\}$ with addition modulo 6 "is" $\Bbb Z_6$. But if there is a sense in which that object "is" $\Bbb Z_6$ and $\{NIL, I, II, III, IV, V\}$ (with suitable addition) "is not" $\Bbb Z_6$, I can't imagine what it would be. But if you can't identify a circumstance under which it would be correct to say that some group "is not" $\Bbb Z_6$ but is "isomorphic to" $\Bbb Z_6$, then it is a distinction without a difference. – 2012-09-28
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0Possibly relevant: [Can skeleta simplify category theory?](http://mathoverflow.net/questions/11674/can-skeleta-simplify-category-theory) – 2012-09-28
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1Actually saying they are equal becomes problematic if you try to compare elements from each group by, say, setting them equal to each other without the intervening isomorphism. This sort of confusion has happened to me when working with several layers of cosets - it's important to remember the isomorphism because without it you end up trying to say that some element is equal to some coset, which doesn't make sense. – 2012-09-28