Determine homomorphism $\displaystyle f:\mathbb{Z}\rightarrow\mathbb{Z}_7$ such that $\displaystyle f(1)=4$.
And then determine $\displaystyle ker(f)\text{ and }f(25)$.
Determine homomorphism $\displaystyle f:\mathbb{Z}\rightarrow\mathbb{Z}_7$ such that $\displaystyle f(1)=4$.
And then determine $\displaystyle ker(f)\text{ and }f(25)$.
Since f is a homomorphism, f(x+ y)= f(x)+ f(y) for all x and y in Z and from that, by induction on n, f(nx)= n f(x) for all x in Z and n a positive integer. Now just calculate:
f(25)= f(25(1))= 25f(1)= 25(4)= 100= 2 (mod 7).
To find the kernel, solve f(n)= f(n(1))= nf(1)= 4n= 0 (mod 7) which is the same as saying that 4n is a multiple of 7: 4n= 7m for some integer m.