Exercise:Show, for $f_i(1_{16})$ and $f_i(x_{16})$, $3$ homomorphism $f_i:\mathbb{Z}_{16}\to\mathbb{Z}_{36},i=1,2,3$ such that $Im f_i\neq Im f_j$ if $i\neq j$.
Case 1) $f_i(1_{36})$. Since both groups are cyclic. I know that $ord\: c_{36}|ord\: 1_{16}$, such that $c_{36}\in\mathbb{Z}_{36}$. So $ord\: c_{36}| 16$. So the homomorphism follow: $f_1(1_{16})=9_{36},f_2(1_{16})=18_{36},f_3(1_{16})=0_{36}$.
Case 2) $f_i(x_{36})$. I do not really understand the $x$ type of questions so my answer was to use the homomorphism found on case 1), in the following way $f_1(x_{16})=9_{36},f_2(x_{16})=18_{36},f_3(x_{16})=0_{36}$.
Question:
1) Are all $f_1(1_{16})=9_{36},f_2(1_{16})=18_{36},f_3(1_{16})=0_{36}$. homomorphism?
2) Did I answer what is intended on case 2)? If not, what is the author asking?
Thanks in advance!