I'm not asking for the formal definition I know it. An isomorphism is a bijective homomorphism. In my book it's indicated many times when two groups are isomorphic, and I don't understand what's the reason for that. What can we "do" when we know that 2 groups are isomorphic? What does it really mean when 2 groups are isomorphic?
Group Theory – Meaning of Isomorphic Groups
abstract-algebragroup-isomorphismgroup-theorysoft-question
Best Answer
It means they are exactly the same except for the names of the elements and the name of the binary operation. An isomorphism between groups is a function that renames all of the elements. (Hence, it is bijective... each element in the first group gets renamed to be exactly one element in the second group.)
The reason we care is that if you are only concerned with the group structure, then the names of the elements or the symbol you use for the binary operation aren't terribly important. Thus, if you know two groups are isomorphic everything about them, in a group theoretic sense, is the same. This is nice since if you can show a group you encounter is isomorphic to a group you already know about, then you get any group-theoretic property of your new group for free.