What Are the Properties of the Group Homomorphism phi from Z(50) to Z(15)?

[Benzoate]

Homework Statement

Suppose that phi : Z(50)->Z(15) is a group homomorphism with phi(7)=6.
a) determine phi(x)
b) Determine the image of phi
c) determine the kernel of phi
d) determine (phi^-1)(3))

Homework Equations

The Attempt at a Solution

I know how to determine phi: I need to find a multiple of 7 where zero is the remainder. the multiple is 13 . Next I would say 7*13=1 mod 15. => phi(13*7)=phi(1) => 6*13= 78 mod 15 =3. There phi(1)=3. Thus, phi(x)=3*x.

I'm not sure how to find the image and kernel of phi. I think in order to determine Ker phi, you say Ker phi={x|3x =1}={1/3)}. My result for Ker phi seems incorrect

 

[zhentil]

Homework Statement

Suppose that phi : Z(50)->Z(15) is a group homomorphism with phi(7)=6.
a) determine phi(x)
b) Determine the image of phi
c) determine the kernel of phi
d) determine (phi^-1)(3))

Homework Equations

The Attempt at a Solution

I know how to determine phi: I need to find a multiple of 7 where zero is the remainder. the multiple is 13 . Next I would say 7*13=1 mod 15. => phi(13*7)=phi(1) => 6*13= 78 mod 15 =3. There phi(1)=3. Thus, phi(x)=3*x.

I'm not sure how to find the image and kernel of phi. I think in order to determine Ker phi, you say Ker phi={x|3x =1}={1/3)}. My result for Ker phi seems incorrect

Are we dealing with Z50 as a group under addition? Your manipulation of phi seems to suggest so, but if so, the identity would not be 1.

What is im(phi)? phi(1)=3, so phi(2) = 6, etc. Does this generate a subgroup in Z15? If so, that's the image. This will also tell you the kernel, since the kernel of a homomorphism is also a subgroup.