2
$\begingroup$

Why are factor groups important?

Thanks for your help

  • 0
    For one, it's a ni$c$e way to create new groups out of existing ones...2012-06-04

1 Answers 1

6

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 $F$.

  • 0
    @JacobSchlather No they don't, $K/E$ is always a Galois extension whenever $K/F$ is.2012-06-04