Help with Isomorphism: Construct an Isomorphism of 25 Elements

[adkinsc]

Can anyone help me with isomorphisms? I am to construct an isomorphism with 25 elements and I am very confused.

Thanks!

 

[Focus]

What sort of isomorphism are we talking about here? Group, ring, field, R-module?

An isomorphism is a bijective homomorphism (it is linear in some sense). When you construct an isomorphism it is usually very simple maps, i.e. from K/I to K would be something like a+I maps to a. If you can be a bit more precise on what you need help on, then I can be more helpful.

 

[adkinsc]

I am not exactly sure on the type. We are working on Boolean Algebra, functions, and inverse functions. My professor gave us a handout that had inverse functions, injective,surjective, and bijective function questions.

The only one I am having problems with is the isomorphism question.

I came up with F:Z₂₅→Z₂₅ and then f:(A)=1, f:(B)=6, f:(C)=8, f:(D)=3, f:(G)=5, f:(H)=2 which equals 25 elements, but I don't think this is close.

 

[Mark44]

I am not exactly sure on the type. We are working on Boolean Algebra, functions, and inverse functions. My professor gave us a handout that had inverse functions, injective,surjective, and bijective function questions.

The only one I am having problems with is the isomorphism question.

I came up with F:Z₂₅→Z₂₅ and then f:(A)=1, f:(B)=6, f:(C)=8, f:(D)=3, f:(G)=5, f:(H)=2 which equals 25 elements, but I don't think this is close.

I understand what this means--F:Z₂₅→Z₂₅--but you haven't specified a rule or formula for your function F, so it's impossible to determine whether it's an isomorphism.

I have no idea what you mean here--f:(A)=1, f:(B)=6, f:(C)=8, f:(D)=3, f:(G)=5, f:(H)=2 which equals 25 elements--what are A, B, C, D, G, and H? And how did you get 25 elements? Do you mean f(A) = 1? You wrote f:(A)=1, which I don't believe means anything.

 

[adkinsc]

Sorry I meant F(A)=1. I added the f(a)=1, f(b)=6, etc. numbers up to get 25. I'm not sure how to specify a rule or formula for my function. I am really confused. What would be the proper way to construct an isomorphism with 25 elements?

 

[Mark44]

Sorry I meant F(A)=1. I added the f(a)=1, f(b)=6, etc. numbers up to get 25. I'm not sure how to specify a rule or formula for my function.

I have absolutely no idea what you're saying here. What function f are you talking about? Are f and F the same thing or different? If they're the same, pick one letter and use it consistently. What are a, b, c, etc.? Is A different from a? What do you mean when you write F(A) = f(a) = 1?

 

[adkinsc]

I am not really sure what I am talking about. I have looked for examples on the internet and ask different people. The project that was given to me by my professor had to do with injective, surjective,bijective, and inverse functions. The isomorphism question stated: Construct an isomorphism containing 25 elements. I don't know anything about this, I'm extremely confused.

I came up with F:Z25→Z25, but I don't know how to define the function or rule.

 

[HallsofIvy]

Okay, they first question, then is "do you know the definition of isomorphism? I would hope so but you haven't given any indication here. Even the way you originally phrased the question makes no sense. We are guessing that you meant "an isomorphism on an algebraic structure (group, ring, field, etc.) with 25 elements" but you don't seem to know what kind of algebraic structure. Z25, I assume, is the cyclic group with 25 elements.

 

[adkinsc]

I know that an isomorphism is a one-to-one correspondence and that everything has to be the same (equal). The class is discrete mathematics and we are working with graph isomorphisms and Boolean Algebra. I know that the question doesn't make any sense but that is how my professor gave it to us on a handout. I have been working on this for a week and still can't find anyone who can help me.