MTH 623, Spring 2007

Brief Notes for Weeks 8-10

Monday, March 19, 2007 (5:00-6:15 PM)

We begin chapter 4, the Sylow theorems, first proving Cauchy’s result about the existence of elements of order p and examining the class equation. The class equation reveals results about the size of the center of a p-group and as a corollary, we show that every group of order p² is abelian. We begin the proof of Theorem 4.6 (which has two typos in the third line: replace G by H in that line.)

Exercise: Show that if G/Z(G) is cyclic then G is abelian.

Wednesday, March 21, 2007 (5:00-6:15 PM)

We prove Theorems 4.6 and Lemma 4.7, then look at the Sylow theorems (without proof) and some of their applications.

Monday, March 26, 2007 (5:00-6:15 PM)

We prove Theorem 4.8, Landau’s Lemma (4.9) and a corollary. Then we look more seriously at the Sylow theorems.

Wednesday, March 28, 2007 (5:00-6:15 PM)

We finish proving some of the main theorems in chapter 4.

We classify all groups of order 21.

Do exercises 4.1-4.4 on page 77; do exercise 4.7 on page 78.

Do exercises 4.11, 4.12, 4.14, 4.15, 4.17 & 4.20 on pages 81-2.

Monday, April 2, 2007 (5:00-6:00 PM)

We do a variety of applications of the Sylow theorems including groups of order 55, or order 2p where p is a prime.

Do exercises on the quaternions: 4.22-4.27 on pages 86-7, then 4.29, 4.30, 4.33, 4.36 & 4.37 on page 87.

Monday, April 2, 2007 (6:00-7:00 PM)

Applications of the Sylow theorems, continued, focusing on the information given by a promised normal subgroup.

We look at groups of order 8.

Last modified April 9, 2007