I started to study group theory but I have many doubts about the topic, so I'd like to share my current understanding together with some questions, my aim is to understand the general ideas and concepts about the topic more than the specific calculations, so answers with little math will be appreciated.
I've seen that a representation of a group is a homomorphism, that means: let $G$ and $H$ be two groups, and $f$ a map from $G$ to $H$ such that $\forall g\in G \Rightarrow f(g)\in H$. Then $f$ is a homomorphism if $\forall g_1,g_2\in G \Rightarrow f(g_1g_2)=f(g_1)f(g_2)$. Now what I understood about this is that a representation conserves the information of the operation inside the group, so $G$ and $H$ may be different but the operations of the two groups follow the same rules. Does it mean that if I know just the operation rules of $G$ and $H$ I can't distinguish them? If yes, can you give me an example of what these abstract rules are?
Then an isomorphism is an homomorphism that is also bijective, which means that if two groups have an isomorphism, one is equivalent to the other and the only way to distinguish them is to look at their elements. So, if a representation would be an isomorphism, I would understand its meaning as another way to express the same group. But, since it's just an homomorphism, I'm struggling in understanding why it's useful to know the representations of a group.