Solve Binary Operations Question: Prove Isomorphism

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SUMMARY

The discussion centers on proving the isomorphism between the structures (Z, .) and (Z, *) using the map φ: Z → Z defined by φ(n) = n + 1. The operation defined as m * n = mn - m - n + 2 is established as the necessary operation for the isomorphism. The user initially struggles with demonstrating that the operations are preserved but ultimately resolves the issue independently.

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  • Understanding of group theory concepts, specifically isomorphisms.
  • Familiarity with the set of integers Z and its operations.
  • Knowledge of function mapping and preservation of operations.
  • Basic algebraic manipulation skills.
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  • Study the properties of isomorphisms in group theory.
  • Learn about operation preservation in mathematical mappings.
  • Explore examples of isomorphic structures in algebra.
  • Investigate the implications of defining new operations on sets.
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Jumblebee
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So I have really been struggling with this question. The original question said: The map [tex]\varphi[/tex]:Z->Z defined by [tex]\varphi[/tex](n)=n+1 for n in Z is one to one and onto Z. For (Z, . ) onto (Z,*) (i am using . for usual multiplication) define * and show that * makes phi into an isomorphism.
I know that the operation must be m*n=mn-m-n+2. But I get stuck in proving that the operations are preserved. When I do [tex]\varphi[/tex](m.n) i get mn+1. and i can't get [tex]\varphi[/tex](m). [tex]\varphi[/tex](n) to work. I think I am doing something wrong. Can anyone help?
 
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Never mind! i just got it to work!
 

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