Group Theory Textbook
From the axioms to the classification of finite groups. A textbook series building definitions, theorems, and proofs in a systematic progression.
Group Theory: A Comprehensive Reference
A single-page overview of undergraduate group theory, from the axioms through the Sylow theorems and the structure of finite abelian groups. Includes key definitions, theorems, and proof sketches with a dependency diagram.
Groups: Definitions and First Properties
Starting from the axiomatic definition of a group, we establish the uniqueness of the identity and inverses, cancellation laws, and the laws of exponents. We then develop the theory of subgroups, cyclic groups, and orders of elements, laying the rigorous foundations for everything that follows.
Cosets and Lagrange's Theorem
Beginning with the definition of cosets of a subgroup, we prove Lagrange's theorem --- the assertion that the order of a subgroup divides the order of the group. As applications, we derive Fermat's little theorem and Euler's theorem, and we exhibit the alternating group A_4 as a counterexample to the converse.
Normal Subgroups and Quotient Groups
We define normal subgroups and establish their equivalent characterizations, then construct the quotient group G/N and prove its basic properties. Topics include the canonical projection, the correspondence between normal subgroups and kernels, subgroups of index 2, and an introduction to simple groups.
The Homomorphism Theorems
We prove the three isomorphism theorems: the first (the image of a homomorphism is isomorphic to the quotient by its kernel), the second (the diamond isomorphism theorem), and the third (cancellation of nested quotients). Each theorem is illustrated with concrete examples and commutative diagrams.
Group Actions and the Class Equation
We introduce group actions and develop the orbit-stabilizer theorem, Burnside's lemma, and the class equation. Applications include the proof that the center of a p-group is nontrivial, the classification of groups of order p^2, the fixed-point theorem for p-groups, and Cayley's theorem.
The Sylow Theorems
We prove Cauchy's theorem on the existence of elements of prime order, then establish the three Sylow theorems on the existence, conjugacy, and number of Sylow p-subgroups. Applications to classifying groups of small order illustrate the power of these results.
Direct and Semidirect Products
We develop the two fundamental ways of building new groups from old: the direct product and the semidirect product. After proving the equivalence of internal and external direct products, we define the semidirect product and illustrate it with the dihedral groups, symmetric groups, and groups of order pq.
The Structure of Finite Abelian Groups
We prove the fundamental theorem of finite abelian groups: every finite abelian group is isomorphic to a direct product of cyclic groups. The two canonical forms --- elementary divisors and invariant factors --- are developed, and the classification procedure is illustrated with explicit examples.
Composition Series and the Jordan--Hölder Theorem
We introduce normal series and composition series, establishing the framework for decomposing a group into simple factors. The Jordan--Hölder theorem proves the uniqueness of the composition factors, and we develop the theory of solvable groups with its connections to the derived series.
Toward the Classification of Finite Groups
We survey the landscape of finite group theory. After stating the classification of finite simple groups, we introduce the sporadic simple groups (including the Monster), discuss the Feit--Thompson theorem, and reflect on what abstract algebra has achieved and what remains open.