find all homomorphisms of Z3 into Z9 find all homomorphisms

Solution

Say ff be the homomorphism and say f(1)=aZ6f(1)=aZ6

in Z3 we have 1+1+1=3=0f(1+1+1)=f(0)=03f(1)=03a=01+1+1=3=0f(1+1+1)=f(0)=03f(1)=03a=0 in Z6Z6 so a=0,2,4a=0,2,4

so you have 3 distinct homomorphism.