G is a cyclic group of order 20 generated by x, G' a cyclic group of order 30 generated by y.

I need to determine if the "function" indicated is well-defined or not...

f: G' to G defined as f(y^i) = x^(i^2+4i+1).

I'm letting y^i = y^j.
Since we're in G', this implies that i=j mod 30.
So 30k = i-j for some k in the integers.

I'm pretty stuck here, this is all that I am pretty sure I'm at least heading in the right direction and saying true things... Any help would be great. Thanks

Let y^i = y^j for some integers i and j,

then i=j mod 30, that is to say : i = j + 30*k for some integer k.

i^2 + 4i + 1 = (j+30k)^2 + 4(j+30k) + 1
i^2 + 4i + 1 = j^2 + 60kj + 900k^2 + 4j + 120k + 1
i^2 + 4i + 1 = j^2 + 4j + 1 + 20*(3kj + 45k^2 + 6k)
i^2 + 4i + 1 = j^2 + 4j + 1 mod 20

and therefore x^(i^2+4i+1) = x^(j^2+4j+1)

hence f(y^i) = f(y^j) : f is well-defined.