Group-Theory on Chen Kai Blog

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

Sun, 19 Sep 2021 09:00:00 +0000

An abstract group is a set with a binary operation satisfying certain axioms. This is elegant but sometimes hard to work with — how do you compute with elements of a group defined by generators and relations, or extract numerical invariants from a multiplication table? The solution, going back to Frobenius and Burnside over a century ago, is to represent group elements as matrices. Matrices are concrete: you can multiply them, take traces, compute determinants, decompose them into eigenspaces. Representation theory is the systematic study of this idea, and it has become one of the most powerful tools in modern algebra, number theory, and mathematical physics.

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

Tue, 07 Sep 2021 09:00:00 +0000

Lagrange’s theorem tells you the order of any subgroup must divide $$|G|$$ . That is a necessary condition, and it is famously not sufficient — the alternating group $$A_4$$ has order $$12$$ , but no subgroup of order $$6$$ . So the moment you start asking “given $$|G| = n$$ , what does $$G$$ actually look like?”, Lagrange leaves you holding an empty bag.

The Sylow theorems are what go inside that bag. They say: for every maximal prime power $$p^a$$ dividing $$|G|$$ , a subgroup of order $$p^a$$ exists, all such subgroups are conjugate, and their count $$n_p$$ is sharply constrained ( $n_p \equiv 1 \pmod p$ , $n_p \mid [G:P]$ ). Ludwig Sylow proved this in 1872, and 150 years later it is still the first thing you reach for when somebody hands you a finite group of unknown order and asks what it is.

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

Sun, 05 Sep 2021 09:00:00 +0000

A group can be enormous — millions of elements, intricate multiplication tables, symmetries that take pages to describe. Yet hidden inside every group are natural compression points where you can collapse entire chunks of the group into single elements, producing a smaller group that still remembers something essential about the original. This article develops the machinery for doing that: normal subgroups, quotient groups, homomorphisms, and the isomorphism theorems that tie everything together.

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

Fri, 03 Sep 2021 09:00:00 +0000

From Abstract Groups to Concrete Actions#

Mental picture before any definitions: a group acting on a set is a collection of moves, and the set is the playground those moves permute. Pick up an object, do a move from the group, set it down. The orbit of an object is everywhere you can take it. The stabilizer is every move that puts it back exactly where it started.

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

Wed, 01 Sep 2021 09:00:00 +0000

Why Algebraic Structure Matters#

Before any definitions, here is the picture I want you to keep in mind. A group is a set in which you can combine any two elements to get a third, undo any element you have produced, and rearrange parentheses without consequence. That is the entire idea, dressed up. The rest of this article is a slow unpacking of that one sentence.