How many kinds of non abelian groups are there?

Question

I know about $S_n$, $D_n$ and $A_n$. And from my limited understanding there seem to be many more. I would like to know whether there is some kind of relation that links a small set of non Abelian groups to create the other ones. Something like with the Abelian groups and the Fundamental Theorem of Abelian Groups.

Answer 1

There is a construction called semi-direct product. It is somewhat like direct product with a twist. This creates a non-abelian group even if the two factors were abelian. And there is a generalisation called group extensions which creates more non-abelian groups.

It is difficult to classify them. Because direct product of two groups with one of them non-abelian will result in a non-abelian group. So one tries to classify simple groups: those not admitting proper non-trivial normal subgroups. Even here it is a vast. Classification Theorem for Simple Groups runs to thousands of journal pages.

The closest analogue of Fundamental Theorem of arithmetic is Jordan-Hölder Theorem for groups. But the same simple ("prime") components can be "put together" in various ways to produce many different non-abelian groups.

If you want some important examples in finite cases: Non-singular matrices of size $n\times n$ (with entries from finite fields).

It has various interesting subgroups, triangular etc.

Answer 2

The "official" answer is the classification of simple finite groups. In some sense, all finite groups are built from simple finite groups, so understanding those is a great help in understanding all finite groups.

However, this is much less tangible and accessible than the classification of finite abelian groups. Perhaps more useful for a beginner is Cayley's theorem, which states that every group is isomorphic to a subgroup of $S_n$ for some $n$. Thus, if you understand all subgroups of $S_n$, you understand all finite groups.

In general, your question "ought" to be difficult to answer; finite groups are very complex objects (as opposed to e.g. finite dimensional vector spaces), and the fact that abelian finite groups are so "easy" to understand tells you that this complexity lies in the non-abelian groups.

Answer 3

In general there is a huge number of ways how to combine certain groups. The relatively most extreme being when the order of the groups are all powers of two.

For example, already for groups with $32$ elements there are $51$ different groups. And it only gets much worse, even for a relatively modest number like $1024$ there are $49487365422$ different groups of that order, and for $2048$ no one knows the exact answer.

Thus while there are unifying construction principles as discussed in other answers what they'll give exactly is hard to predict.