Math Help - group homomorphism

1. group homomorphism

24. Suppose that ƒ:Z50 to Z15 (co-domain is Z 15) is a group homomorphism with ƒ(7)=6.
a. Determine ƒ(x).
b. Determine ƒ(Z50).
c. Determine ker ƒ.
d. Determine ƒ–1({3}).

I am lost particularly on b, a,c and d I think I will be able to get by myself. Any suggestions to help me for b?
Thanks

2. Originally Posted by nhk
24. Suppose that ƒ:Z50 to Z15 (co-domain is Z 15) is a group homomorphism with ƒ(7)=6.
a. Determine ƒ(x).
b. Determine ƒ(Z50).
c. Determine ker ƒ.
d. Determine ƒ–1({3}).

I am lost particularly on b, a,c and d I think I will be able to get by myself. Any suggestions to help me for b?
Thanks

First, as $(7,50)=1$ then $<7>=\mathbb{Z}_{50}$ and then $\forall\,x\in\mathbb{Z}_{50}\,\,\exists\,n\in\math bb{Z}\,\,\,s.t.\,\,\,7n=x$ $\Longrightarrow f(x)=f(7n)=nf(7)=6n\!\!\!\pmod{15}$.

Since $ord(6)=5\,\,\,in\,\,\,\mathbb{Z}_{15}$ , you get that $f\left(\mathbb{Z}_{50}\right)=<6>\cong \mathbb{Z}_5$ .

I did two, now you do the other two.

Tonio