Let $G=\left \{ ax^{2}+bx+c \mid a,b,c \in \mathbb{Z}_{3} \right \}$, Add elements of G as you would polynomials with integer coefficients, except use addition modulo 3. Prove that G is Isomorphic to $\mathbb{Z}_{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{3}$
Suppose $g_{1}=a_{1}x^{2}+b_{1}x+c_{1} , g_{2}=a_{2}x^{2}+b_{2}x+c_{2}$
Define:
$\phi: G \rightarrow \mathbb{Z}_{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{3}$
$g \mapsto \left ( a,b,c \right ) $
Now,
$\phi\left ( g_{1}+g_{2} \right )=\phi\left ( \left ( a_{1}+a_{2} \right )x^{2}+\left ( b_{1}+b_{2} \right )x+\left ( c_{1}+c_{2} \right ) \right ) =\left ( a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2} \right ) =\left ( a_{1},b_{1},c_{1} \right )+\left ( a_{2},b_{2},c_{2} \right ) =\phi\left ( g_{1} \right )+\phi\left ( g_{2} \right )$
Is my attempt in the right direction?
Thanks in advance.
Yes, that is already a canonical isomorphism and I don't think much else interesting about it. Using the addition operator then
$$\phi(ax^2 + bx + c) + \phi(dx^2 + ex + f) = (a, b, c) + (d, e, f) = (a + d, b + e, c + f) \text{ mod } 3 = \phi((a+d)x^2 + (b + e)x + (c + f))$$
It is evidently onto and one-to-one, you are done.