Groups

Chapter 1: Intro. to groups
- dihedral groups

Chapter 2: Groups
- Examples: Z/nZ, matrix groups, etc., see table 2.1, p. 41
- elementary properties of groups

Chapter 3: Finite groups; Subgroups
- order of a group, of an element
- subgroup tests
- the center of a group; the centralizer of an element

Chapter 4: Cyclic groups
Let G = <a^k> be a cyclic group of order n.
- a^i = a^j iff i = j mod n
- a^i is a generator for G iff gcd(a,n) = 1
- Each subgroup of G is cyclic and its order divides n. For each divisor, k, 
  of n there is exactly one subgroup of G of order k: <a^(n/k)>.  
- The number of elements of order d in G is phi(d) = |U(d)|.

Chapter 5: Permutation groups
- Definition of the symmetric group, cycle notation.
- Basic properties of cycles:
  -- every cycle can be written as the product of disjoint cycles
  -- disjoint cycles commute
  -- every cycle is the product of 2-cycles
- The alternating group.

Chapter 6: Isomorphisms
- basic properties of isomorphisms
- every group is isomorphic to a permutation group
- automorphisms

Chapter 7: External direct products
- definitions
- the order of an element in a direct product is the lcm of the orders
  of the components
- Z/(n_1 ... n_k)Z = Z/n_1Z x ... x Z/n_kZ iff the n_i's are pairwise 
  relatively prime
- structure of U(n) 

Chapter 8: Cosets and Lagrange's theorem
- basic properties of cosets
- Lagrange's theorem: the order of a subgroup divides the order of the
  group 
- stabilizers and orbits

Chapter 9: Normal subgroups and factor groups
- if a subgroup is normal, its set of cosets forms a group in a
  natural way

Chapter 10: Homomorphisms
- basic properties
- the kernel of a homomorphism is a normal subgroup and vice versa