Sufficient conditions for a group homomorphism/isomorphism

Question

So I was wondering what are sufficient conditions $\phi$ has to fulfill to be a group homomorphism/isomorphism.

I know that when we consider cyclic groups, $\phi(x)$ is an isomorphism $\iff$ you the generator to another generator (Obviously the groups need to have the same order). Now for homomorphism, we just have to map the generator to an element whose order divides the order of the generator and then we have a homomorphism.

However, things get more complicated when we consider non-cyclic groups, or at least I think. I know what are necessary conditions for a homomorphism/isomorphism are, but not what sufficient conditions are. So I was wondering if there are any theorems regarding this.

Answer 1

Given a group $G$ and a group $H$.

We need a couple of definitions, first

Definition: We say that a subset $S\subseteq G$ is a generating set for $G$ if the only subgroup of $G$ which contains $S$ is $G$ itself.

Observation: The set of all elements in $G$ is a generating set for $G$, so every group has a generating set.

Now we also need the notaion of "relations".

Definition: Given a generating set $S$ of a group $G$. We say that a word $s_1s_2...s_k$ where $\forall_i s_i\in S$ is a relation if $s_1\cdot s_2\cdot...\cdot s_k = 1_G$.

Given a group $G$ and a generating set $S$. In order to define a homomorphism $\varphi:G\rightarrow H$. It suffices to send each element in $S$ to an element in $H$ in a way which preserves relations. That is, let $h_i = \varphi(s_i)$ then for any relation $s_1s_2...s_k$ it must satisfy that $h_1\cdot h_2\cdot...\cdot h_k = 1_H$.

The same with isomorphism, you need to send each element in the generating set of $G$ to an element in the generating set of $H$ which preserves relations.