Review and study pointers for Exam 3

Our last hour test  will cover the material in Chap.3.5 and 3.6 (Group action on a set, orbits, applications of G-sets to counting ), 4.1 (isomorphism theorems) and 4.2 (p-Groups). You will not be responsible for any material in the text but not covered in class such as simple groups  commutator subgroups and series of groups.

There are three basic areas:

• Definitions (approximately 25%)
• Computations and examples (approximately 40%)
• Proofs of Theorems or Lemmas covered in class or other simple facts (approximately 35%)

Definitions

• Group action and G-set
• Orbit
• Xg and G_x
• Join of subgroups HvK
• Center of a group
• Normalizer of a subgroup
• Simple group
• p-Group

Computations and examples

• Be able to calculate X_g and G_x for various G-sets X.
• Be able to to use Burnside's formula to calculate the number of orbits for various counting problems.
• Be able to use the first and second isomorphism  theorems. For example use the first isomorphism theorem show that A_n is a normal subroup of S_n such that S_n/A_n contains two elements.
• Be able to apply the theory of p-groups and the class equation to given groups.

Proofs and statements of theorems

You will not be responsible for the proof of Burnside's Formula (3.6.1) or Cauchy's Theorem (4.2.3). However, you have to be able to state and understand what the theorems say, in order to be able to use it. Be able state and to reproduce the proofs of:

• The Second and Third Isomorhism Theorems (Theorems 4.1.5 and 4.1.7)
• Lemma 4.1.4
• Theorem 4.2.1

In addition, I may ask you to give proofs of various simple facts that you may or may not have seen in class.

Good Luck. If you have questions ask in class, see me or send me e-mail at ebarbut@uidaho.edu.