{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

BONUS NOTES

# Algebra

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 150a: Modern Algebra Cross-sections and complements Since the book does not much discuss cross-sections 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 cross-section 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 cross-section that I am describing is a left cross-section (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 cross-section doesn’t have to be a right cross-section.) There are five degrees of goodness of cross-sections 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

{[ snackBarMessage ]}

### Page1 / 2

BONUS NOTES - Math 150a Modern Algebra Cross-sections and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online