Is there a simple way to determine the number of homomorphisms of a group $G$ generated by a finite subset $A$ in $Z_m$?
I was studying and I came across that if $A$ generates the group $G$ the amount of homomorphism of $G$ in $\mathbb{Z}_m$ is $m^{|A|}$. I don't understand why. What do you think: $m$ is the number of elements of $\mathbb{Z}_m$, so does $m^{|A|}$ correspond to a homomorphism of a single element?
Ps: if instead of homomorphism it was just a map, can we say that the number of maps of $G$ in $Z_m$ is $m^{|A|}$?
If $G$ is the free group generated by $A$, then by the universal property, the number of homomorphisms from $G$ to $\Bbb Z_m$ is $m^{\lvert A\rvert}$.
As for maps in general, not necessarily homomorphisms, from any $G$ to $\Bbb Z_m$, we get $m^{\lvert G\rvert }$ such.