This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 150a: Modern Algebra Crosssections and complements Since the book does not much discuss crosssections and complements, here are some notes that may be helpful. This topic is related to sections 2.8 and 2.10 of the book. If G is a group and A ⊆ G is a subgroup, then the left cosets { gA } of A are a partition of G . As a set, they are called G / A , the left coset space of A . A subset B ⊆ G is a crosssection of G / A if B has one element in each left coset of A . If B is also a subgroup of G , then it is a complement of A . More precisely, the kind of crosssection that I am describing is a left crosssection (because it cuts across left cosets), and if it is a subgroup, a left complement. However, a left complement is also a right complement (exercise GK4(a)). (A left crosssection doesn’t have to be a right crosssection.) There are five degrees of goodness of crosssections and complements. 1. If A is not normal and B is not a subgroup, then G / A is not a group either. In this case you can still say that G / A ∼ = B as sets, and G ∼ = A × B as sets. This is a way to say thatas sets....
View
Full Document
 Spring '03
 Kuperberg
 Algebra, Cyclic group, Coset, cosets, left complement

Click to edit the document details