Mathematical Parenthesis on Groups and Lagrange Theorem

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Lagrange's Theorem: Applications and Homomorphisms

Covers Lagrange's theorem, group homomorphisms, congruence classes, and normal subgroups.

Quotients in Abelian Groups

Covers quotients in abelian groups and the concept of free abelian groups, showing that every abelian group is isomorphic to a quotient of a free abelian group.

Shor's Factoring Algorithm

Covers Shor's factoring algorithm, aiming to find integer factors efficiently using quantum computation.

Symmetries in Regular Polygons

Delves into the symmetries of regular polygons, including the dihedral group and left cosets in a finite group.

Subsets and subgroups associated with an action

Explores subsets, subgroups, actions, fixed points, stabilizers, orbits, and bijections in group theory.

Number Theory: GCD and LCM

Covers GCD, LCM, and the Euclidean algorithm for efficient computation.

Cohomology: Cross Product

Explores cohomology and the cross product, demonstrating its application in group actions like conjugation.

Number Theory: Greatest Common Divisor and Prime Factorization

Introduces greatest common divisor, prime factorization, and the Euclidean Algorithm.

Algebra: Practice Exam Solutions

Covers the solution of a practice exam in Algebra, focusing on finding greatest common divisors of polynomials and exploring group properties.

Bar Construction: Homology Groups and Classifying Space

Covers the bar construction method, homology groups, classifying space, and the Hopf formula.