1. Isomorphism

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?

2. In one word it is HUGE!

3. Originally Posted by Also sprach Zarathustra
In one word it is HUGE!
Can you be a bit more explicit - maybe a few categories or examples?

4. Originally Posted by Also sprach Zarathustra
Interesting.

I'm writing another article on magic squares and I found an isomorphism between certain classes of magic squares which may be of interest to some people (the article will come out next year).

5. Originally Posted by wonderboy1953
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?
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, $M$, is a homomorphism between vector spaces. If it, as a matrix, has non-zero determinant it is an isomorphism.

Example 2: $\mathbb{Z}$ and $2\mathbb{Z} = \{2i: i \in \mathbb{Z}\}$ are isomorphic as groups, $\phi: 1 \mapsto 2$, and in general $\phi: i \mapsto 2i$, defines an isomorphism. This is an isomorphism as it is clearly a bijection, and because,

in groups a map $\varphi$ is a homomorphism if $(a \ast_1 b)\varphi = a\varphi \ast_2 b\varphi$ where $\ast_1$ and $\ast_2$ are themultiplications' in the respective groups. Here the `multiplication' is addition,

$(i+j)\phi = 2(i+j) = 2i+2j = i\phi+j\phi$ as required.

6. Originally Posted by Swlabr
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.
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.

7. Originally Posted by HallsofIvy
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).