# Thread: group homomorphism

1. ## group homomorphism

24. Suppose that :Z50 to Z15 (co-domain is Z 15) is a group homomorphism with (7)=6.
a. Determine (x).
b. Determine (Z50).
c. Determine ker .
d. Determine 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 :Z50 to Z15 (co-domain is Z 15) is a group homomorphism with (7)=6.
a. Determine (x).
b. Determine (Z50).
c. Determine ker .
d. Determine 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 $\displaystyle (7,50)=1$ then $\displaystyle <7>=\mathbb{Z}_{50}$ and then $\displaystyle \forall\,x\in\mathbb{Z}_{50}\,\,\exists\,n\in\math bb{Z}\,\,\,s.t.\,\,\,7n=x$ $\displaystyle \Longrightarrow f(x)=f(7n)=nf(7)=6n\!\!\!\pmod{15}$.

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

I did two, now you do the other two.

Tonio