Math 5331 Schedule/Homework
 

 

Date Topic Homework Due date Reading assignment
Tuesday 3/4 Group actions Study the examples given in class.
Study the actions given on  p.43 (#4),  p. 52 (#1, 2)
  pp. 41-45, 51-53
Thursday 3/6 Transitive group actions Study the actions given on p.113 (#2, 3, 4) and on      p.115    (# 4, 5).   pp.112-117
Tuesday 3/11 Cayley's Theorem Do problems # 4, 5, 6, 13, 14, 15, 16, 18 on pages 44-45 3/27
HW # 7
 
pp. 118-121
Thursday 3/13 The class equation Do problem # 8 on page 53.
Handout
II on Groups
4/1
HW # 8
pp.122-129
March
17-21

Spring Break 

Spring Break 

 

Spring Break 

Tuesday 3/25 No class. Work on Handout II.     pp. 133-144.
Thursday 3/27 The Sylow Theorems Study the following:
      Examples on page 144
      Propositions 21 and 23
      Corollary 22 on page 145.
  pp. 144-146
Tuesday  4/1 The Sylow Theorems
Take home test (Test # 3 given out.)
Due Tuesday April 8.
Handout III on Groups.
 
  pp. 152-165
Thursday 4/3 Direct product.
Finitely generated abelian groups.
p.  147 # 13, 18, 30
pp. 156-15 7
# 1, 18 a-c
pp.165-166 # 1 a, b; 2 a-c; 3 a-c; 4
4/15
HW # 9
pp. 188-194
Tuesday 4/8 Abelian groups.
Nilpotent groups.
Handout IV on Nilpotent Groups   pp. 194-199
Thursday 4/10 Solvable groups Problem # 8 on page 198.
Handout V on Solvable groups.
pp. 173-174 # 1, 2, 4, 7, 10
4/22
HW # 10
pp. 223-230
Tuesday 4/15 Introduction to Rings pp. 230-231 # 1, 2, 3, 4, 7, 9, 11, 15, 21    
Thursday 4/17 Ring homomorphisms.
Ideals
p. 237 # 1
p.249 # 16, 18, 22, 24a, 27, 28
   
Tuesday 4/22 Review      
Thursday 4/24 Test # 4      
Tuesday 4/29        
Thursday 5/1        
Tuesday 5/6 Final Exam
11:00 am-1:30 pm
     

 

 

 

Date Topic Homework Due date Reading assignment
Thursday 2/7 Cosets      
Tuesday 2/12 Normality. Quotient groups. Prove the following:
1.  Corollaries to Lagrange's theorem.
2. Prove the equivalent definitions of normal subgroup.
3. If , then .
4. If A and B are normal subgroups of G, then their intersection is also normal in G.
5. If N is a normal subgroup of G and
, then .
2/21
HW # 4
 
pp. 89-100
Thursday 2/14 The isomorphism theorems 1. pp. 85-88 # 3, 4, 22, 24, 30, 31, 36
2. pp. 95 # 1, 4, 5, 8
3. Study the statements and proofs of Propositions 13, 14, and 15 on pages 93-94.

4. Study the proof of the Proposition given in class.
2/21
HW # 4
pp. 101-105
Tuesday 2/19 Simple groups Handout I on Groups -ALL 2/26
HW # 5
pp. 106-110
Thursday 2/21 Alternating groups 1. Show that if G is a simple group, then any homomorphic image of G is either isomorphic to G or of order one.
2. p111 # 1, 2, 3
3/6
HW # 6
 
Tuesday 2/26 Examples      
Thursday 2/28 Test # 2      

 

 

 

 

Date Topic Homework Due date Reading assignment
Tuesday 1/15 Introduction to groups: Definitions and Examples
  1. pp 21-23 # 1,2, 6, 9a
  2. Study the proof of Proposition 2 (p.20)
1/24
HW # 1
pp 16-25; 29-32
Thursday 1/17 Properties of groups.
Dihedral and Symmetric groups
  1. Prove: Sn is a non-abelian group for all n greater than or equal to 3.
  2. pp. 21-23 # 11, 18, 20, 22, 23, 25, 29,31, 32, 34
  3. pp. 32-33 # 1, 4, 6, 7
  4. p.36 # 1
1/24
HW # 1
pp 36-39; 46-48
Tuesday 1/22 Homomorphisms and isomorphisms
  1. Prove: The normalizer of a subset A of G is a subgroup of G.
  2. Prove: The centralizer of a subset A of G is a subgroup of G.
  3. Prove: Isomorphism of groups is an equivalence relations.
  4. pp. 39-41 # 1, 2, 3, 4, 8, 17
    pp.48-49 # 10 a
1/31
HW # 2
pp 49-51, 54-56
Thursday 1/24 Normalizers and centralizers.
Cyclic groups
  1. Prove directly that Z(G) is a subgroup.
  2. Prove Claim 3 from class.
  3. pp 52-53 # 3, 5 (a, b), 6
  4. Study the proof of Theorem 4 on p. 56
1/31
HW # 2
pp 56-64
Tuesday 1/29 Generators
  1. Study the proof of Propositions 5, 6 on p. 57.
  2. Compute <a> for all all a in Z/36Z.
  3. Study the proof of  Theorem 7 on p. 58.
  4. If H and K are subgroups of a group G, is HUK a subgroups of G? Prove or give a counterexample.
  5. pp 60-61 # 1, 2, 3, 4, 10, 11
  6. p. 65 # 1, 5, 6
2/7
 
HW # 3
pp 66-71
Thursday 1/31 Lattices of subgroups  p. 71 # 2 (a, c, d), 9 (a, b)  2/7
HW # 3
 
Tuesday 2/5 Test # 1