What is the no. of injective group homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ ( when $m \le n$)? What is the no. of injective group homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ ( when $m \le n$) ? I know it should not exceed $\gcd(m,n)$ , but I cannot calculate the exact number . Please help. Thanks in advance 

The image of an injective homomorphism $C_m \to C_n$ must be a subgroup of order $m$.
By Lagrange's theorem, this is only possible when $m$ divides $n$, and in this case there is only one possible image because $C_n$ contains exactly one subgroup of each order dividing $n$.
The question then reduces to where we can send the generator of $C_m$. 
We must send it to an element of order $m$ in $C_n$. There are exacty $\phi(m)$ such elements.
Therefore, there are $\phi(m)$ injective homomorphism $C_m \to C_n$ when $m$ divides $n$ and none otherwise.
($\phi$ here is Euler's totient function.)
