Do isomorphic groups have to have the same order?

Question

I have been doing a lot of practice problems for an Abstract Algebra course, and I see a lot of proofs using the fact that isomorphic groups have the same order, which intuitively makes sense to me since they must have the same structure/are onto. But, I was wondering if this fact comes from a specific theorem, or if it is a result of something stated in a theorem/properties of isomorphic groups? Any ideas? (Or is it as simple as the fact that it is a bijection?)

Answer 1

Yes, because it is a bijection between groups. So Cardinality of isomorphic groups are equal.

Answer 2

Isomorphic groups are, to all intents and purposes, exactly the same. You will often heard it said that two things are the same "up to isomorphism", for this reason.

In particular, the fact that an isomorphism is a bijection does give you the particular fact that you're after, but you should really try to conceptualise an isomorphism as a simple "relabelling" of things.

For example, consider the additive group of integers modulo $2$ (i.e. $\mathbb{Z}_2$), and the multiplicative group of non-zero integers modulo $3$. These groups are isomorphic (can you see why?), but if I actually forget what things are called, I simply have the 2 element group, which is unique. You can call it $\mathbb{Z}_2$ or anything else, it's still the same group (up to isomorphism).