HOMEWORK
Assignment 1 - Herstein
- Page 8. 3,4a,8-12
- Page 16. 13,15
- Page 23. 3,5,6,7
Assignment 2
- Page 35. 1,3-5, 8, 10, 12, 14, 21, 23, 24
- Page 80. 1-3, 10
Assignment 3
- Page 47. 2-6, 9, 12-18, 27, 29
- construct multiplication table for D(4), the dihedral group of 8 elements preserving the square
Assignment 4
- Page 53. 2,3,5,9,10,11,12,16,17,18,21
(note in problem 17, there is a misprint, namely i=0,1)
- Optional (not to be handed in): Page 53. 15,20
Assignment 5
- Page 64. 2-6,9,11,17. Optional: 16,20
- Page 70. 2,4-6. Optional: 7,8
NOTE:
- First Mid-semester exam: Monday, October 16.
- In-class, closed book, no notes
- Homework will be collected October 16 not October 13.
Assignment 6 (due Monday October 23)
- Page 65. 15
- Page 70. 7,8,9,10
- Page 74. 3 (read 2 first), 5
- Page 80. 11,12,13
Assignment 7
- Page 90. 2,4,8-12,14,16-18
Assignment 8
- Page 102. 8,10,11a,13,14
- Prove a group of order 33 must be cyclic.
- Prove that a group of order 21 is either cyclic or is
isomorphic to the group described in Herstein on
page 69 with generators a of order 7 and x of order 3
satisfying ax=xa^2 (here a^2 means a squared).
- Page 108. 3-6.
Assignment 9
- Page 103. 17
- How many elements of order 7 are there in a simple group of order 168.
- Let G be an infinite group in which every element has finite p-power order
p a prime number. Prove that either G has a subgroup of order p^n for
n, n > 0, or there exists a natural number N such that every finite subgrup of G has order bounded by N
- If P is a normal p-Sylow subgroup of a finite group G, and f is a
homomorphism from G to itself (i.e, an endomorphism) then f(P) is
contained in P.
- If H is a normal subgroup of order p^k of a finite group G, then H is
contained in every p-Sylow subgroup of G.
- P.108. 7,9,10,16,18
- Optional: Let G be a finite group with order O(G) = (p^n)q where p
and q are distinct
primes and p^n means "p raised to the nth power". Assume p>q. Show
G has a unique subgroup of index q.
- Optional: Prove: every group of order 200 must contain a normal Sylow
subgroup
Assignment 10
- Page 130. 4, 7-9, 12
- Page 135. 2,3,5-10
Assignment 11
- Page 136. 11-13, 18, 21
- Page 149. 4,7
- Page 158. 1-4, 6
Additional problems that might be useful
- Page 142. 4,5,6
- Page 166. 5, more to come
- More: Page 167. 8,9,10,11,12,13