Here is the question:

Suppose that ϕ: Z_50 to Z_15 is a group homomorphism with ϕ(7) = 6

a)determine ϕ(x)

b) determine the image of ϕ

c) determine the kernel of ϕ

d) determine ϕ^(-1)(3). That is, determine the set of all elements that map to 3.

--- a) doesn't ϕ(x) = (6/7)x

b) ok, so the image of ϕ is something that i am confused on, we didn't really talk about in class and was barely covered in my book. My professor told me that the image of ϕ is denoted phi(G) and is the set of all elements of G-bar that are 'hit' by ϕ. so does this mean that the image of ϕ is any element in Z_15 such that that element times 6/7 gets me an element in Z_50?

c)I understand the kernel. It is any element in Z_50 times 6/7 that gets me the element, that is 0 in Z_15 (an element of 15). but there are not many of these, only {0, 35} i think this is right, but i am a little skeptical about ϕ(x)=6/7

d)so this is similar to the kernel, but instead of mapping to the identity, 0, it maps to three? if that is the case then i know that 21 is an example, but otherwise i am having a difficult time. thank you for any help.