Group-Theory

Sep 19, 2021 Abstract Algebra 62 min read

Abstract Algebra (10): Representation Theory — Groups Acting on Vector Spaces

Representing abstract groups as matrices makes them concrete and computable — Maschke's theorem, Schur's lemma, and character theory give us powerful classification tools.

Sep 7, 2021 Abstract Algebra 66 min read

Abstract Algebra (4): Sylow Theorems — Dissecting Finite Groups

The Sylow theorems give us a systematic way to find and count subgroups of prime-power order — the sharpest tool for classifying finite groups.

Sep 5, 2021 Abstract Algebra 70 min read

Abstract Algebra (3): Quotient Groups and Homomorphisms: The Art of Collapsing Structure

Normal subgroups, quotient constructions, and the isomorphism theorems — how to systematically simplify groups while preserving their essence.

Sep 3, 2021 Abstract Algebra 68 min read

Abstract Algebra (2): Group Actions — How Groups Move Things Around

We formalize how groups act on sets, prove the orbit-stabilizer theorem, derive Burnside's lemma, and count necklaces.

Sep 1, 2021 Abstract Algebra 70 min read

Abstract Algebra (1): Groups — Your First Encounter with Algebraic Structure

From integers to symmetries, we build the formal definition of a group, prove Lagrange's theorem, and compute our first subgroup lattice.