TITLE: The most common theorems taught in Abstract Algebra
QUESTION [12 upvotes]: I am self learning abstract algebra. I want to know which theorems are a must to understand. 
Now these are limits I have to deal with (please consider when answering): 


I have limited internet access
Few mathematical books written in English are available. 
I can not afford to order books from abroad．　


I just want to know what is the core knowledge (theorems, lemmas, etc) of any decent graduate level abstract algebra class.

REPLY [15 votes]: Assuming you are aiming for a Groups/Rings/Fields, at a level one might expect from someone who has taken a year's undergraduate sequence but not much more:
Groups

Definition, lots of examples.
Subgroups, homomorphisms, normal subgroups 
Cosets; Lagrange's Theorem.
Group actions.
Permutation groups. Symmetric and alternating groups.
Cayley's Theorem.
The Isomorphism Theorems.
Cauchy's Theorem and the Sylow Theorems.
Abelian groups.
The Fundamental Theorem of Finitely Generated Abelian Groups.

Rings

Definition, lots of examples.
Homomorphisms.
Ideals.
The Isomorphism Theorems.
Field of quotients of integral domains.
Polynomial rings.
Euclidean rings, Principal Ideal Domains, Unique Factorization Domains
Gauss's Lemma and polynomials over Unique Factorization Domains.

Fields

Field extensions.
Algebraic extensions. Dedekind's Product Theorem.
Primitive Element Theorem.
Separability.
Fundamental Theorem of Galois Theory (finite case).
Solvability by radicals.
Finite fields.

I think Herstein's Topics in Algebra is a good resource, though it is a bit old-fashioned (it has almost nothing on group actions, for example). 
With a more time, I would add the following topics:
Groups

Introduction to Semigroups.
Semidirect products; extensions.
Jordan-Höder Theorem.
Commutator subgroup; solvability (this will have been covered somewhat when 
discussing solvability by radicals under Fields; study it in more detail).
Basic results on $p$-groups.
Divisible and reduced abelian groups.
Structure Theorem of Divisible Abelian Groups.
Free groups, free products, free products with amalgamation.
Basics of representation theory of groups.

Rings

Jacobson radical.
Wedderburn's Theorem.
Wedderburn-Artin Theory.
Division rings.
Localizations.
Group rings.
Basic theory of modules.
Modules over PIDs.

Fields

Inseparable extensions.
Transcendental extensions.
Kummer Theory.
Normal Basis theorem.
Fundamental theorem of Galois Theory (infinite case).

Lang's Algebra is a good resource, but is not easy to learn from. For Ring Theory, Lam's First Course in Noncommutative Ring Theory is very good.