Binary Structures and Mapping

I am having great difficulty conceptualizing binary structures and mapping from one to the other.

Here's my problem:

The map defined by for is one-to-one and onto . Give the definition of a binary operation on such that is an isomorphism mapping.

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So I have . This means that for members a and b, this structure is and , right? And I'm trying to map this to an unknown structure? I don't understand what exactly I'm doing.

What is that function? I assume it takes a number in and maps it to a number in , specifically one more than the input number. But what does that have to do with my structures? What am I trying to find?

In utter confusion, I took and figured that for the inputs of a=2 and b=3, the answer would be 6 since 2*3=6. So I take that 6... add one to it from , and that gives me 7... is that 7 the value that is supposed to correspond with the second structure, ? So is my task to find a function that takes two variables and to make 7, or more abstractly, ab+1?? (Headbang)

Any help = appreciated.

Re: Binary Structures and Mapping

what you are being asked to do is FIND a definition for * so that φ(ab) = φ(a)*φ(b).

the fact that φ is 1-1 and onto already is the "iso" part....ensuring that φ(ab) = φ(a)*φ(b) is the "morphism" part.

suppose that a*b is defined as ab+1...does this definition of * fit the requirements?

(your question is a little vague...since no other information is given, i am assuming you aren't requiring that * be associative, or have an identity).