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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 Euler--Hamilton divide: why "all edges" and "all vertices" are worlds apart
Determining whether a graph has an Eulerian circuit (traversing every edge exactly once) reduces to checking vertex degrees. Determining whether it has a Hamiltonian cycle (visiting every vertex exactly once) is NP-complete. This article explains why one word makes all the difference.
Jordan Normal Form: Beyond Diagonalization
Every complex square matrix is similar to a Jordan normal form — a block-diagonal matrix of Jordan blocks that is unique up to block ordering. We prove existence and uniqueness, connect the Jordan form to the minimal polynomial, and apply it to compute the matrix exponential e^{tA}.
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Cosets and Lagrange's Theorem
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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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The Euler--Hamilton divide: why "all edges" and "all vertices" are worlds apart
Determining whether a graph has an Eulerian circuit (traversing every edge exactly once) reduces to checking vertex degrees. Determining whether it has a Hamiltonian cycle (visiting every vertex exactly once) is NP-complete. This article explains why one word makes all the difference.
The Homomorphism Theorems
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