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So, most of us have been in an introductory algebra course and proved basic facts about isomorphic groups (or rings, modules, etc., we'll use groups as the example and call them $G$ and $H$), such as

"$G$ and $H$ have the same number of elements of order $n$ for any $n\in\mathbb{N}$", or "$G$ is abelian if and only if $H$ is abelian", and so on.

After a while, we stop bothering to prove that properties we care about are always preserved by isomorphism. For instance, we'd find it very natural to say that if $G$ is isomorphic to a nilpotent group then $G$ is nilpotent, and use it without hesitation in a proof. And indeed, we're never failed because isomorphisms do indeed encode the "sameness" of two groups.

But I'm wondering if there is still some rigorous way of saying that these types of properties will always be shared? I'm not doubting that this is true, but it's just something I was curious about.

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  • $\begingroup$ "After a while, we stop bothering to prove that properties we care about are always preserved by isomorphism." We don't prove them, but we should mentally that everything seems OK. In any case of (even tiny) doubt, we should prove it. I am (re)proving very basic set theory facts because I am not 200% sure of them. E.g.: if $f\circ g$ is one-to-one, which of $f$ or $g$ has to be one-to-one? Never been able to memorize it... so I prove it 4 or 5 times a year. $\endgroup$ – Taladris May 13 '16 at 0:02
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    $\begingroup$ You might wish to look at universal properties from category theory which is one way of formalizing the idea of properties that are preserved over isomorphism. $\endgroup$ – Q the Platypus May 13 '16 at 0:22
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    $\begingroup$ There are cases when not all properties of the group are preserved. For example, it is not necessarily the case that isomorphic groups given in terms of permutations have the same action on the underlying set. More formally, they may be isomorphic as groups but not as permutation groups (see Wikipedia for more details). I should stress, however, that this is about a group given with an action rather than just the group. $\endgroup$ – user1729 May 13 '16 at 12:44
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It depends on what you mean by "properties." One sort of tautological definition is "a property is something invariant under isomorphism," which makes the statement you want tautologically true.

A less tautological definition is "a property is a first-order statement in the first-order language of groups," and then it's a theorem that if two groups are isomorphic then the same first-order statements are true in them (elementary equivalence). But many natural examples of properties are not expressible as first-order statements, e.g. simplicity. So you can move on to second-order statements, and it's again a theorem that if two groups are isomorphic then the same second-order statements are true in them.

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    $\begingroup$ This is exactly the type of response I wanted. I thought it might have to do with mathematical logic, although I know nothing about that entire area $\endgroup$ – Alex Mathers May 12 '16 at 23:53

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