Structure Theorem for non-abelian finite groups or rings How many structure theorems do we have in Abstract Algebra for finite algebraic structures? I know some of the following theorems:

*

*If $G$ is a finite abelian group, then $G$ is a product cyclic groups of prime power order.

*If $R$ is a finite commutative ring, then $R$ is Artinian and hence a direct product of local rings.

I am wondering if there are other structure theorems for finite algebraic structures.
Is it possible to write a non-commutative or a non-abelian group in this way?
Why are the theorems available for abelian cases but not for non-abelian ones?
If someone can please help me find some references where I can get structure theorems for non-commutative or a non-abelian group/rings, I will be thankful.
Please help.
 A: Non-abelian finite groups are quite complicated, but we have completely classified the finite simple groups. Here's an overview from Wikipedia. Moreover, every finite group can be "built out of finite simple groups" in an essentially unique way, in the same sense that every nonzero integer can be written as a product of primes in an essentially unique way! If you want to know precisely what this means, this is the content of the Jordan–Hölder theorem (you will first need to learn what a composition series is). In contrast to prime factorizations, where we just multiply the primes to reproduce the original integer, there are many complicated ways that simple groups can be put together to form a non-simple group. The general problem of understanding the possible isomorphism types of a group from its simple factors is what's known as an "extension problem". In short, while we know all the finite simple groups, and we know  that all finite groups are built from these, we are not close to understanding all finite groups.
Non-commutative finite rings are quite complex! Wedderburn's little theorem says that a finite ring with no zero divisors is a field. Besides this, I don't know of any general classification results for finite rings. Even the finite commutative rings are quite difficult to classify: we easily reduce to a product of finite local rings, but those are complicated to describe. The good news is that, by considering the possible underlying abelian groups, it is not too computationally expensive to compute all isomorphism classes of finite commutative local rings of any given cardinality $n$.
I also feel obligated to mention the $5/8$ theorem:

Theorem ($5/8$ Theorem for Finite Groups) Let $G$ be a finite group such that the probability that two randomly chosen elements of $G$ commute is greater than $5/8$. Then $G$ is abelian.
Theorem ($5/8$ Theorem for Finite Rings) Let $G$ be a finite ring such that the probability that two randomly chosen elements of $G$ commute is greater than $5/8$. Then $G$ is commutative.

See here for a proof, or here for a generalization to compact topological groups.
