Homomorphism and Isomorphism importance

Question

From group theory, two groups $(G,\cdot)$ and $(S,*)$ are homomorphic if there is a map $f$ such that $f(a\cdot b)=f(a)*f(b)$. While these groups are isomorphic if the map $f$ is homomorphism and bijective. I have two fundamental questions.

1. If $G$ is only homomorphic to $S$, what properties do they have common? E.g. if $G$ is abelian whether $S$ will be abelian? If $G$ is cyclic whether $S$ will cyclic? If $G$ has $n$-subgroups whether $S$ has $n$-subgroups?

2. What properties of $G$ and $S$ will have in common if they are isomorphic?

Answer 1

A group is an algebraic structure, a set together with a binary operation.

A homomorphism is a special kind of map between two groups because homomorphism respect the group operation.

An isomorphism is a bijective map which maintain the equivalence of the underlying sets and a homomorphic nature preserve the group operation. Two group are isomorphic mean those groups have structural similarities. As the group theoretic point of view, they are not different at all. i.e two groups have same group theoretic properties (cyclic, abelian, order, order of elements, no of elements of same order etc) . i.e if $G\cong G'$ by the isomorphism $\phi$ then we can view the elements of $G'$ as a relabeling of the elements of $G$ everything else are same(in group theoretic point of view)

For the case of homomorphism the properties are preserved onto the image of the homomorphism.

$\phi : G\to G'$ homomorphism then -

1. $G$ abelian implies $\phi(G) $ abelian.

2. $G=\langle a\rangle$ implies $\phi(G)=\langle \phi{(a)}\rangle$

3. $|a|, \phi(a) <\infty$ implies $|\phi(a) |$divides $|a|$ etc.