A question on group isomorphism.
Let (A,・) and (B,*) be some groups, and I want to show they're isomorphic. I know there has to exist a bijection function f: Α → Β such that f(x・y)=f(x)*f(y) for all x,y in A. I'm wondering if there can be multiple such bijection functions and it is sufficient to find just one such function in order to prove A and B are isomorphic. If there can be multiple of them, could you provide an example of that case?
Another question is if there exist multiple isomorphisms, is the structural similarity between A and B the same with different isomorphisms (meaning what kind of isomorphism picked does not affect how the structural similarity between A and B are preserved)?
Thank you in advance!
Best Answer
Welcome to MSE!
Yes. There might be multiple bijections. As a simple example, look at
$$\mathbb{Z} / 2 \times \mathbb{Z} / 2 \cong \mathbb{Z} / 2 \times \mathbb{Z} / 2$$
We can use the identity, which is an isomorphism. We can also use the flippy isomorphism $(a,b) \mapsto (b,a)$ (which is its own inverse).
As for your second question, I'm not sure what you mean by whether the "kind of isomorphism picked does not affect how the structural similarity between $A$ and $B$ [is] preserved"...
I will say that no matter what isomorphism you use, all the group-theoretic properties will be the same. For instance, their orders, how many subgroups, whether there exist torsion elements, etc. So I'll say that as a good first answer: no, it doesn't matter at all which isomorphism you use.
Of course, like everything in math, this is only mostly true. In fact, different isomorphisms carry different information, and just because two groups are isomorphic doesn't always mean we want to consider them the same! A good example of this is the study of lattices in $\mathbb{R}^n$. These are all (abstractly) isomorphic to $\mathbb{Z}^n$, but it turns out to be a bad idea to identify them all as "the same".
There are also times when different isomorphisms carry different information. A famous example is the isomorphism of a (finite dimensional) vector space to its double dual. There are lots of ways to do this (abstractly) but it turns out there's one unique best choice of isomorphism. To formulate this properly we need the language of category theory, and I'm not sure if this is the right place to go into that. Rest assured, though, you'll hear more about it as you continue progressing in mathematics!
The long and the short of it is that every isomorphism will preserve all the group theoretic properties of your object of interest. However different choices of isomorphism can sometimes lead to counter intuitive statements (galois theory is rather infamous for this...). If there is a canonical way to make the isomorphism, rather than finding some random isomorphism by making arbitrary choices along the way (choosing a basis, etc.) that canonical isomorphism tends to be "better behaved" in a way that you'll learn about when you start learning about categories.
I hope this helps ^_^