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Cosets and quotient groups: the art of controlled forgetting
The quotient group G/N is the first major conceptual hurdle in group theory. By computing small examples by hand -- from clock arithmetic to D4 modulo its center -- we build the intuition for what "dividing" a group really means, and why normality is indispensable.
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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Popular Articles
Cosets and quotient groups: the art of controlled forgetting
The quotient group G/N is the first major conceptual hurdle in group theory. By computing small examples by hand -- from clock arithmetic to D4 modulo its center -- we build the intuition for what "dividing" a group really means, and why normality is indispensable.
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.
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