MATH 3360-001 20 October 2010
Answer the problems on separate paper. You do not need to rewrite the problem statements on your answer sheets. Do your own work. Show all relevant steps which lead to your solutions.
For no question is True/Yes or False/No by itself sufficient as the answer. Retain this question sheet for your records.
1.
Definitions
Exam II
Make-up
2. (8 pts) Given an example of a group with the indicated combination of properties a.
b.
c.
d.
finite cyclic group with exactly 10 generators infinite Abelian group that is not cyclic finite Abelian group that is not cyclic finite non-abelian group of order more than 8
3. (14 pts) Let and let a. (4 pts) b. (4 pts) c. (6 pts)

= 

1 2 3 4 5 6 7 8
2 7 8 3 4 6 1 5




= 

1 2 3 4 5 6 7 8
4 7 6 2 5 3 8 1



Calculate
Calculate
2
and
Write
and
as products of disjoint cycles and calculate |
4. (12 pts) Determine which of the follow subsets of S
7 are subgroups of S
7
?
c.
d.
a.
b.
H =
1
{(13 7), (3 7), (17 3), (1)}
H
2
= {(2 4), (2 4 7), (2 7 4), (2 4)(2 47), (2 4)(2 7 4), (1)}
H
3
= {(2 4), (15 7), (17 5), (2 4)(15 7), (2 4)(17 5), (1)}
H =
4
{(2 4), (3 6), (2 4)(3 6), (1)}
5. (12 pts) Identify the cosets of < 12 > in ℤ
30
.
6. (12 pts) Let H = { φ ∈ S
7
φ = 1 and φ (6) = 6} . Find the index of H in S
7
.
7. (14 pts) Let H be a subgroup of G . For that ~ is an equivalence relation on G .
,
G , let a ~ b if and only if ab
− 1
H . Prove

8. (8 pts) Let G be a group such that | G | 1500 . Suppose that H is a subgroup of G such that
| H | 54 and K is a subgroup of G such that | K | 60 . What are the possible orders of G ?
9.(12 pts) Find the remainder of 21 2669 upon division by 23.