Verifying Solutions to Isomorphism Problem: Need Help!

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SUMMARY

The discussion focuses on verifying solutions to the isomorphism problem, specifically parts (a) and (b). The user, Alexis87, received feedback indicating that their proof for part (a) was circular and needed revision. For part (b), the recommendation was to demonstrate the distinctness of images of the mapping function f to establish its injectivity, thereby confirming its bijectivity due to equal cardinality of the sets involved.

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  • Understanding of isomorphism in algebraic structures
  • Familiarity with bijective functions and their properties
  • Knowledge of modular arithmetic, specifically $[a]_6$, $[a]_2$, and $[a]_3$
  • Experience with mathematical proofs and logical reasoning
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  • Learn about modular arithmetic and its applications in isomorphism proofs
  • Explore techniques for constructing mathematical proofs without circular reasoning
  • Investigate alternative methods for proving isomorphisms in algebraic structures
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Mathematicians, students studying abstract algebra, and anyone interested in understanding isomorphism proofs and their verification processes.

Joe20
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Hi, I have attached the question and the solutions to part a and b of this question. Would like someone to verify if I have done anything wrong. Greatly appreciate it! Thanks.

Would also like to check if there is a simpler method to prove f is an isomorphism? Thanks
 

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Hi Alexis87,

Your proof for part (a) is somewhat circular — the goal was to prove $[a]_6 = _6$ implies $([a]_2,[a]_3) = (_2, _3)$, but you started with that claim. Remove it.

For part (b), remove the beginning part of your proof. To show that $f$ is bijective, you could show that the images $f(j)$, $j = 0,\ldots, 5$, are distinct, and so $f$ is injective. Since $f$ is an injective mapping between two sets of the same cardinality, it is also surjective. Hence, $f$ is bijective.
 

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