- #1
oneGirlArmy
- 4
- 0
Homework Statement
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]], [S6, [tex]\circ[/tex]]
c. [[tex]Z[/tex]2, +2]], [S2, [tex]\circ[/tex]]
Homework 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.
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 S2 = {(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.
Last edited: