(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am suppose to determine if the following list of groups are isomorphic and if they are define an isomorphic function for them.

a. [5[tex]Z[/tex], +],[12[tex]Z[/tex], +] where n[tex]Z[/tex] = {nz | z[tex]\in[/tex][tex]Z[/tex]}

b. [[tex]Z[/tex]_{6}, +_{6}]], [S_{6}, [tex]\circ[/tex]]

c. [[tex]Z[/tex]_{2}, +_{2}]], [S_{2}, [tex]\circ[/tex]]

2. Relevant equations

+_{6}means x +_{6}] y = the remainder of (x+y)/6

To prove not isomorphic we are suppose to show that the two sets are not one-to-one, or one is commutative while the other is not, etc.

3. The attempt at a solution

For a, I am fairly certain they are isomorphic and that the function should be f(x) = (12/5)x since it is a bijective function and f(x+y) = f(x) + f(y).

For b, My gut feeling is that it is not isomorphic however I can't find a good reason why. Perhaps because the second group is not commutative. However that answer just doesn't sit well with me.

Finally, for c I am confused because S_{2}= {(1,2), (2,1)} while [tex]Z[/tex]_{2}={0, 1,2} so it seems like there could be an isomorphic function but I'm uncertain what that function could be without it being piecewise for each element 0, 1, and 2.

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# Homework Help: Abstract Alg- Group theory and isomorphic sets.

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