A Quick Refresher Guide for Group Theory

Part I — Foundations

1. Group — the definition and axioms. Symmetry stripped to essentials.
2. Order — |G| and |a|. Why finiteness matters.
3. Subgroup — subsets closed under the operation.
4. Lagrange's theorem — the order of a subgroup divides the order of the group.
5. Homomorphism — maps that preserve the operation.
6. Normal subgroup — subgroups stable under conjugation.
7. Characteristic subgroup — subgroups stable under every automorphism.
8. Hall subgroup — order coprime to index; Schur–Zassenhaus and semidirect products.
9. Quotient group — the isomorphism theorems.
10. Group action — orbit-stabilizer and Burnside's lemma.
11. Conjugacy class — the class equation.
12. Center — elements that commute with everything.
13. Commutator subgroup — the "most non-abelian" part.
14. Abelian group — fundamental theorem of finite abelian groups.

Part II — Structure of finite groups

1. Direct product — the simplest construction.
2. Semidirect product — direct product with a twist.
3. Sylow theorems — existence and counting of prime-power subgroups.
4. p-group — groups of prime-power order.
5. Derived series — iterated commutator subgroups.
6. Solvable group — derived series reaches {e}.
7. Burnside's pᵃqᵇ theorem — two primes force solvability.
8. Upper central series — climbing out from the center.
9. Lower central series — descending via commutators with G.
10. Nilpotent group — either series terminates.
11. Fitting subgroup — the canonical normal nilpotent piece.
12. Frattini subgroup — non-generators and the elementary abelian quotient.
13. Extension — fitting two groups into a short exact sequence.
14. Simple group — the atoms of classification.
15. Chief series — the canonical structural fingerprint.

Part III - Presentation And Classification

Coming soon...

Part IV - Finite Group Computation Algorithms

Coming soon...