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Combinatorial Designs and Latin Squares: Balanced Arrangements
We define Latin squares and prove their existence, introduce mutually orthogonal Latin squares (MOLS), develop the theory of balanced incomplete block designs (BIBDs), prove Fisher's inequality, and discuss the connection with finite projective planes.
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From the axioms to the classification of finite groups. A textbook series building definitions, theorems, and proofs in a systematic progression.
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From the axioms of vector spaces to Jordan normal form. A textbook series building definitions, theorems, and proofs in a systematic progression.
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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.
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.
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.
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