## Tags » Group Theory

#### A theorem of Schur on commutator subgroup

The aim of this note is to prove the following theorem, due to Schur, and to exhibit some corollaries. The proof given here comes from J. 763 more words

#### Structure Theory of Modules over PIDs, Part IV: Jordan Canonical Form

In the last post, I proved the spectral theorem for modules over PIDs and applied it to show the Chinese Remainder Theorem and a more general form of the linear algebraic spectral theorem. 1,113 more words

#### Positive curvature versus negative curvature

A leitmotiv in geometric group theory is to make a group act on a geometric space in order to link algebraic properties of with geometric properties of . 661 more words

#### Structure Theory of Modules over PIDs, Part III: Spectral Theory

In the last post, I introduced the ideas of irreducibility and length, which gave us the basic framework to discuss module structure theory. I also placed additional constraints on the rings by making them PIDs, which makes them easier to work with in what follows. 1,523 more words

#### Structure Theory of Modules over PIDs, Part II: Modules over Rings

In the last post, I presented a series of eerily similar results in linear algebra and finite abelian group theory and suggested that these were both specific cases of a more general structure theory of modules over rings. 1,849 more words

#### Structure Theory of Modules over PIDs, Part I: Motivation

In this series of posts I will explore the structure theory of modules over PIDs, a beautiful and useful generalization of standard linear algebra and finite abelian group theory (and one of the most beautiful parts of Math 55a). 862 more words