$\quad$ Finding the number of homomorphisms from group $G$ to a group $H$, if both are cyclic and have same order, then we have to map the generators of both the groups, this leaves us the possibility and the number of possible elements is the number of homomorphisms. What if one of them is not cyclic? or not even abelian? then of course the element $1$, and its image $a$ ($f(1)=a$) the order of $a$ should divide both the order of $H$ and order of $1$. This leaves the possible elements in $H$ one of them to be $a$, resulting in number of homomorphisms.
I have few questions
- At First, am I right in grasping the homomorphism concept?
- We all know that each homomorphism helps in creating Normal Subgroup (Kernel), so counting the number of normal subgroups of each order, can we get the number of homomorphisms?.
I read the articles in mathstack regarding the group homomorphisms but still I cant catch the monkey...