MTH 623, Spring 2007

Brief Notes for Weeks 11-13

(very tentative)

Wednesday, April 4, 2007 (5:00-6:00 PM & 6:00-7:00 PM)

We begin chapter 5, studying normal series and group extensions (chapters 5 & 7 of Rotman’s textbook.)

“Factoring” groups: extending K by G/K. (More generally, extending K by Q.)

Example: C₁₁ by C₅ gives all groups of order 55.

This provides a recursive (or inductive) process, so we introduce the concept of normal series.

This leads to the concept of composition series of a group (but is this concept “well-defined”?)

The best possible normal series would be one in which each factor is abelian; thus the composition series consists of factors which are cyclic of prime order)

This leads to solvable series and the concept of solvable group.

Examples of some nice normal series [Remember – abelian is easy!]

Ascending central series (using the center to go up.)

A group is nilpotent if the ascending central series works! (That is, yields a normal series.)

Derived series = descending central series (using the commutator subgroup to go down.)

The derived series works if and only if the group is solvable.

Monday, April 9, 2007 (5:00-6:15 PM)

We continue looking at normal series. Matrix groups provide an enlightening example.

(What would you guess is the commutator subgroup of GL(n,K)?

What would you guess is the center?)

The concept of “meta”.

For a property P, we might consider meta-P groups, where both K and G/K have property P. Eg. metacyclic, metabelian.

Theorem. Meta-solvable implies solvable.

Characteristic subgroups.

Characteristic is transitive, that is, H char K, K char G implies H char G.

(Furthermore, if K is normal in G then so is H.)

Similarly, one could define “fully invariant” subgroups. (These are mapped into themselves by all group homomorphisms.)

Galois & the relationship with solvable. (See notes on "What Galois Saw".....)

Wednesday, April 11, 2007 (5:00-6:15 PM)

We finish up the brief introduction to Galois theory.

We skim chapter 6, studying some basic results on direct products, in preparation for semidirect products.

Monday, April 16, 2007 (5:00-6:15 PM)

We look at the semidirect product of K by Q. We assume K is normal and G = KQ and the intersection of K and Q is just {1}. Then every element q of Q acts as an automorphism of K and so we may use the automorphism group of K to build G.

Wednesday, April 18, 2007 (5:00-6:15 PM)

We continue the study of semidirect products.

Last modified April 9, 2007