5520,  Fall 2004: Review Sheet for Midterm Exam

The test will be in roughly the following format: there will be two parts, one on theory and one on computation.  Each part will contain 4 or 5 problems, and you will have to do three of them.  Here are brief outlines of all the sections we have covered, along with numerous review problems.  There are probably too many review problems for you to do all of them, so I suggest you read over them and try to figure out solutions to those you are not sure how to do.  Many of them won't take too long.

Chapter 0:  Know the division algorithm,
                  Theorem 2 (GCD(a,b) is a linear combination of a and b),
                  and Euclid's Lemma (p | ab implies p | a or p | b). 
                  Be comfortable with arithmetic in Z_n and induction arguments. 
                  Everything in the section on functions is prerequisite material.
                  Problems:  13, 26, 27

Chapter 1:  Be able to compute in D_n, especially D_4.
                  Problems:  10, 12, 16

Chapter 2:  Definition of groups, basic examples and properties (Theorems 1-3).
                  Problems:  8, 13, 24, 30, 36 (use Chapter 4 for 36)

Chapter 3:  Order, subgroups. 
                  Theorems 1-3 tell you how to decide if a given subset is a subgroup. 
                  Know <a>, Z(G), and C_a(G).
                  Problems:  7, 11, 13, 24, 27, 34, 36, 52-54 (52-54 are proofs: good practice)

Chapter 4:  Cyclic groups: this section is very important. 
                  Theorem 1: when a^i = a^j. 
                  Its corollaries: |a| = |<a>|, and a^k = e iff |a| divides k.
                  Theorem 2: In Z_n, <a^k> = <a^d> for d = GCD(k,n).
                  Its corollaries: when <a^i> = <a^j>, which a^i generate <a>, which i generate Z_n.
                  Theorem 3: Complete description of the subgroups of cyclic groups.  (Crucial!).
                  Theorem 4: If d | n, the number of elelments of order d
                  in a cyclic group of order n is phi(d).
                  Its cor.: phi(d) divides the number of elements of order d in any group.  (Useful!)
                  Problems:  7, 24, 25, 32, 34, 39, 44, 50, 55, 56, 58, 59, 63, 64

Chapter 1-4 Supplemental Problems: 4, 7, 12 (use Chapter 7), 18, 21, 26, 28, 34, 36

Chapter 5:  Permutation groups.  Know 2-line and cycle notation.
                  Given x in S_n, be able to find x as a product of disjoint cycles,
                  and as a product of 2-cycles.
                  Be able to compute products, inverses, and orders in S_n.
                  Know what A_n is and how to decide if a cycle is odd or even.
                  Problems:  8, 21, 23, 31, 43, 45, 49

Chapter 6:  Isomorphisms.  Know their definition.  We'll skip Theorem 1.
                  Theorems 2-3: properties of isomorphisms.
                  Be able to prove whether or not two groups are isomorphic.
                  Know what Inn(G) and Aut(G) are.
                  Thm 5: Aut(Z_n) = U_n
                  Problems:  9, 19-21, 40-42

Chapter 7:  Cosets: know their properties.  We covered up through Corollary 5.
                  Lagrange's Theorem: if H a subgroup of G, then |H| divides |G|.
                  Corollaries: (1)  The number of cosets is |G| / |H|.
                                    (2)  |a| divides |G| for all a in G.
                                    (3)  |G| prime implies G cyclic.
                                    (4)  a^|G| = e for all a in G.
                                    (5)  For all a in Z_p, a^p = a.
                  Problems:  4, 9, 13, 23, 30, 34