Is there a rigorous way of saying, “if $G$ and $H$ are isomorphic then $G$ and $H$ share all the same properties”?

Question

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.

Answer 1

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.