In one word it is HUGE!
M-W defines isomorphism as: "...a one-to-one correspondence between two mathematical sets; especially : a homomorphism that is one-to-one..."
Can somebody explain what the significance of this concept is to math or science? Why do we need to know about isomorphisms?
Graph Isomorphism -- from Wolfram MathWorld
for instance...
I can't think of any example of an isomorphism that isn't also a homomorphism. You'd just call it a bijection then (bijection = one-to-one correspondence).
So, an isomorphism is a bijective homomorphism. But what is a homomorphism? It is a structure-preserving mapping; your magic squares/groups/rings/etc. have a structure and you want to preserve it.
Basically, an isomorphism is a mapping which preserves structure and size. So it comes down to one word: labelling. Two magic squares/groups/rings/etc. are isomorphic if you can relabel one to get the other. Whenever you read `isomorphic' you should read it as `the same' or `identical'.
For example, write a magic square using roman numerals. Why is that different from using 1, 2, 3... ? Well, it's not. The concept of an isomorphism formalises this.
Example 1: A linear map, , is a homomorphism between vector spaces. If it, as a matrix, has non-zero determinant it is an isomorphism.
Example 2: and are isomorphic as groups, , and in general , defines an isomorphism. This is an isomorphism as it is clearly a bijection, and because,
in groups a map is a homomorphism if where and are the`multiplications' in the respective groups. Here the `multiplication' is addition,
as required.
The usual definition of "isomorphism" is that it is an invertible homomorphism. The first definition wonderboy1957 gave, " one-to-one correspondence between two mathematical sets", is much more general and is normally just called a "one-to-one" correspondence. Wonderboy1957, an isomorphism between two mathematical systems essentially says they are the same. If you just changed the "names" of the objects in on system, you would have the other.
I can assure you there's no name changing, but a process that shows that two different-looking magic squares having a one-to-one correspondence. There's more I can say, but please wait for my article (btw please update your records to reflect that my handle is Wonderboy1953).