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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.
Combinatorics: A Complete Summary of Definitions, Theorems, and Proofs
A single-page survey of undergraduate combinatorics. Covers the fundamentals of counting, binomial coefficients, inclusion-exclusion, generating functions, Catalan numbers, Ramsey theory, Burnside and Polya enumeration, combinatorial designs, posets, and algebraic combinatorics. Includes a dependency diagram and a theorem-algorithm directory.
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.
Popular Series
Group Theory Textbook
From the axioms to the classification of finite groups. A textbook series building definitions, theorems, and proofs in a systematic progression.
Graph Theory Textbook
From the definition of a graph to planarity, coloring, and matroids. A textbook series building definitions, theorems, and proofs in a systematic progression.
Number Theory Textbook
From divisibility and congruences to p-adic numbers and algebraic integers. A textbook series building definitions, theorems, and proofs in a systematic progression.
Combinatorics Textbook
From the fundamentals of counting to algebraic combinatorics. A textbook series building definitions, theorems, and proofs in a systematic progression.
Popular Articles
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.
Burnside's Lemma in Action: 57 Essentially Different Colorings of a Cube
How many essentially different ways can you paint the six faces of a cube using three colors? Burnside's lemma gives the answer: 57. We derive this by classifying the 24 rotations of the cube and counting fixed colorings for each, building from orbits and stabilizers to the general counting formula.
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.
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.
The Spectral Theorem: Orthogonal Diagonalization of Symmetric Matrices
Every real symmetric matrix can be orthogonally diagonalized — we prove this spectral theorem and its complex generalization for normal operators. The spectral decomposition A = sum lambda_i P_i into orthogonal projections is derived, and we apply it to classify quadratic forms via Sylvester's law of inertia.
Combinatorics meets dynamic programming: when counting becomes computation
Dynamic programming is bottom-up evaluation of a recurrence; a generating function is its analytic closed form. We draw the correspondence between the knapsack DP and generating function multiplication, discuss convolution and FFT, and survey the connections across the entire series.
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