Brief notes for Week 1 (tentative):

MTH 623, Spring 2007

Brief Notes for Weeks 1-3

Monday, January 8 (5:00-6:15 PM)

This material is (officially, at least) review material. We will look at Chapter 1 of Rotman’s textbook, Groups and [group] homomorphisms.

We review permutations (including cycle notation and permutation parity.) Note that we will assume the function fg is g first, followed by f, using the so-called “left-handed” notation (of undergraduate mathematics.)

We look at properties of permuations – these form the core ideas form the definition of a group.

Some examples of groups: the group of units of a ring. GL(n,k), etc. (p. 13.)

We review homomorphisms. Note that “basic” results of the definition, appearing in Lemma 1.13, p. 17.

Look at the following problems:

page 5: 1.13,

page 7: 1.17, 18,

page 10: 1.19, 20, 21,

page 15: 1.26, 29, 33, 35, 37,

=> I will set aside time from 6:15 to 6:45 after class as “open office hours” to go over the above problems and any other early questions you might have.

Wednesday, January 10 (5:00-6:15 PM)

We review homomorphisms.

We review subgroups, including cyclic subgroups. We define order of an element (p. 21.)

A group homomorphism always gives two important subgroups, the kernel and the image; we first touch on the Fundamental Homomorphism Theorem.

We prove some results about cosets and introduce the index of a subgroup.

We prove Lagrange’s theorem.

We look at applications of Lagrange’s theorem (lines in Euclidean geometry, cosets of integers, subgroups of S_n)

(Check out: the dihedral group in color.)

We look at some applications of Lagrange’s theorem (Fermat’s Little Theorem and RSA.)

Look at the following problems:

pages 18-19: 1.46, 48, 49.

page 23: 2.2, 7, 8, 9,

page 27: 2:11, 12, 20, 23,

page 29: 2.24.

=> I will set aside time from 6:15 to 6:45 after class as open office hours to go over the above problems and any other early questions.

Monday, January 15 (No class, in honor of Dr. Martin Luther King)

Wednesday, January 17 (No class)

Monday, January 22 (5:00-6:15 PM)

We review normal subgroups. Normal subgroups are a critical concept for this course. (Why? Why do we need this concept of “normal”?)

Note the product formula on page 30. (“Never underestimate a formula which counts something.”)

We discover how to multiply cosets! (When we are able to multiply cosets, we have a group of cosets – this is the “quotient group” or “factor group”.)

Note the definition of commutator, page 33. (Weird, right? Completely invisible in a commutative world, the commutator is very important in the noncommutative environment.)

We go through the isomorphism theorems.

Look at the following problems:

page 31: 2.25, 26, 30, 31, 34-37, 39.

=> I will set aside time from 6:15 to 6:45 after class as open office hours to go over the above problems and any other early questions.

Monday, January 22 (6:45-7:45 PM, second session)

We review the isomorphism theorems, including the correspondence theorem and prove the FHT, the Fundamental Homomorphism Theorem. (Mathematicians use the phrase “fundamental theorem” in a special way – one usually cannot go further in the subject without understanding “the fundamental theorem”.)

The FHT will touch every class lecture, in some way, after this.

Look at the following problems:

page 39: 2.53, 59,

=> I will also take some more questions on homework or work some of these problems at 7:45, for 15 minutes or longer, if desired.

Wednesday, January 24 (5:00-6:15 PM)

Applications of the FHT.

Direct Products.

Look at the following problems:

page 40: 2.62,

page 42: 2.78.

=> I will set aside time from 6:15 to 6:45after class as open office hours to go over the book problems, assignment problems and any other early questions.

Wednesday, January 24 (6:45-7:45 PM, second session)