**5520, Fall 2004: Review Sheet for Midterm
Exam**

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