I know a bit about simple groups. A finite Abelian group is a (direct) product of finite cyclic groups. The simple finite Abelian groups are exactly $\mathbb{Z}_p$ for $p$ a prime. And so, I understand how all finite Abelian groups are made up of finite simple groups.
But, from what I understand, all finite groups are in some way made up of (finite) simple groups.
My question is: how does that work? What does it (more precisely) mean that a finite group is made up of simple groups?
Edit: Thanks to Stefan for directing me to questions that have basically already have the answer. I have done a bit more of research on this and I think I can narrow my question a bit. I would like to understand how simple groups are the building blocks of all finite groups. That is, I would like to understand how, given a finite group $G$, one can find (or show there exists) simple groups $G_1, \dots, G_n$ such that $G$ is [insert something] of $G_1,\dots G_n$. From here, I understand now that it somehow has to do with composition series and Jordan-Holder's Theorem. I think I understand the definition of a short exact sequence. From that same question:
Then $G$ is built from some uniquely determined (Jordan-Hölder) simple groups $H_i$ by taking extensions of these groups.
I still don't get how this group $G$ is determined by the simple groups.
I guess I am looking for more details basically putting together how one starts with a finite group $G$, "finds" simple groups $G_1, \dots G_n$ and then says that $G$ is isomorphic to something in terms of the simple groups.
Best Answer
Every finite group could be decomposed as a finite number of extensions of simple groups. What is meant by extension could be read here; this is a different form as for example to decompose solely as a direct product (but a direct product could be seen as a special case of a group extension). By repeated extensions by simple groups you get a so called composition series, and these are unique up to a permutation of their composition factors (the quotient groups of successive groups in a series) by Jördan-Hölder. If you are interested there is also a theory for decompositions based on direct products, see Krull-Schmidt theorem.
This question appeared here already in a more or less different flavour, take a look at these posts and their answers:
How is a group made up of simple groups
The "architecture" of a finite group