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Math Help - Isomorphism

  1. #1
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    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?
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    In one word it is HUGE!
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    Quote Originally Posted by Also sprach Zarathustra View Post
    In one word it is HUGE!
    Can you be a bit more explicit - maybe a few categories or examples?
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    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).
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by wonderboy1953 View Post
    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 the`multiplications' in the respective groups. Here the `multiplication' is addition,

    (i+j)\phi = 2(i+j) = 2i+2j = i\phi+j\phi as required.
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    Quote Originally Posted by Swlabr View Post
    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.
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  8. #8
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    Quote Originally Posted by HallsofIvy View Post
    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).
    Last edited by wonderboy1953; December 6th 2010 at 08:07 AM. Reason: clarifiying
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