Is it possible to translate theorems between isomorphic groups?

[Fredrik]

The definition of isomorphism says that the two structures are the same up to relabelling. Tbqh I don't see what other explanation you're seeking for the fact that they can be "considered equivalent for all practical purposes". Surely if we have a two-element group consisting of elements a,b and a two-element group consisting of elements c,d, then we don't need any additional explanation for why they're the same.

I understand what you're thinking, but I'm not so sure I agree with it. You're thinking that this is like when we use a Riemann integral to make the idea of "area under the graph" precise. It's not like the area is something else, and the integral just a technique to calculate it. The integral defines what we mean by "area" in this context. Similarly, the concept of "isomorphism" makes another idea precise. The thing is, I would say that the idea it makes precise is the idea of "relabeling", not the idea of "equivalent". It seems to me that there should be a way to make that idea precise and then prove that it's equivalent to "isomorphic". If the idea is "equivalent for all practical purposes", the mathematical concept that makes that idea precise should specify exactly what the "practical purposes" are.

 

[mathwonk]

Here is a statement true of G but not always of H: "The elements of the group are elements of G".

so I think you should be a little careful by the range of the word "theorem".