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.