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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 doesnt 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....
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This note was uploaded on 02/01/2008 for the course MATH 150A taught by Professor Kuperberg during the Spring '03 term at UC Davis.
 Spring '03
 Kuperberg
 Algebra

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