Focal subgroup theorem

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Group Theory Fundamentals

Introduces group theory basics and group action concepts with associated subsets.

Cohomology Groups: Hopf Formula

Explores the Hopf formula in cohomology groups, emphasizing the 4-term exact sequence and its implications.

Symmetries in Physics

Explores symmetries in physics through group theory, emphasizing how symmetries keep equations of motion unchanged.

McKay Correspondence and Coxeter Groups

Explores the McKay correspondence, Coxeter groups, and finite subgroups of SU(2) and SO(3, emphasizing odd order properties and root system constructions.

Mathematical Parenthesis on Groups and Lagrange Theorem

Explores cosets in commutative groups, Lagrange theorem, and integer factorization.

Isomorphism Theorems: Third Isomorphism Theorem

Explores the third isomorphism theorem in group theory, focusing on quotient groups and a categorical perspective.

The Degree of a Covering Space

Shows the equivalence between the index of a covering space's fundamental group and the degree of the covering space.

Hidden Sub-group Problem and Simon Algorithm

Explores the hidden sub-group problem and Simon algorithm in quantum computing.

Symmetries in Regular Polygons

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

Homomorphisms: Birational Maps and Affine Varieties

Covers homomorphisms between affine varieties, birational maps, regular groups, and connectedness.