Why are factor groups important?

Question

Why are factor groups important?

Thanks for your help

Answer 1

Among many other reasons:

1. Factor groups of $G$ capture all possible images of $G$ under homomorphisms (via the Isomorphism Theorems);

2. Factor groups of $G$ are "smaller" than $G$, so one may hope that they may be simpler to study than $G$ itself. The Isomorphism Theorems allow us to "import" a certain amount of information from factor groups to the groups themselves.

3. If you think of groups in terms of Galois Theory, with $G$ being the Galois group of a Galois field extension $K/F$, then the factor groups of $G$ corresponds precisely to the Galois groups of subextensions $E/F$, $E\subseteq K$, that are Galois over $F$.