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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.
Catalan Numbers and Lattice Paths: The Reflection Principle and Bijective Proofs
We prove the formula C_n = (1/(n+1)) C(2n, n) for the Catalan numbers using the reflection principle, state the Lindstr\"{o}m--Gessel--Viennot lemma, construct bijections among five combinatorial interpretations of the Catalan numbers, solve the ballot problem, and introduce Narayana numbers.
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.
Popular Series
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.
Group Theory Textbook
From the axioms to the classification of finite groups. A textbook series building definitions, theorems, and proofs in a systematic progression.
Linear Algebra Textbook
From the axioms of vector spaces to Jordan normal form. 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.
Directed Graphs and Topological Sorting
Strongly connected components (SCCs), Tarjan's and Kosaraju's algorithms, DAGs and topological sorting, the correspondence with partial orders, and dynamic programming on DAGs.
Trees and Forests: The Minimal Connected Structures
We prove the equivalent characterizations of trees, establish the formula $|E|=|V|-1$, introduce rooted trees and Cayley's formula, and develop the theory of spanning trees and minimum spanning trees via Kruskal's and Prim's algorithms.
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.
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.
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