Math 667
Discussion Questions #2
September 7, 2010
8. A ring is a structure consisting of a non-empty set R, two binary operations and , and two
elements named 0, 1 R so that
(a) Properties of
For
For
For
For
For
any
any
any
any
any
x, y R, we have x y R
Math 667
Homework Questions #3
Due: September 28, 2010
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
10. Show that if a divides b and a divides c,
Math 667
Homework Questions #4
Due: October 5, 2010
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
14. Consider Theorem 3 from the text. Show that
Math 667
Homework Questions #5
Due: October 12, 2010
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
17. Let R be a commutative ring, and consider t
Math 667
Homework Questions #6
Due: October 26, 2010
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
21. Write down the multiplication table for the
Math 667
Homework Questions #7
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
26. Find all of the generators of the groups R7 and R8 .
27. Is D3 cy
Math 667
Homework Questions #8
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
34. Let
M =
cos
sin
sin
cos
and
M =
cos
sin
sin
.
cos
What
Math 667
Homework Questions #9
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
37. Show that if a and b are elements of a group G, then (ab)1 = b1 a
Math 667
Homework Questions #10
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
42. Prove that if G and H are isomorphic and G is abelian, then so i
Math 667
Homework Questions #2
Due: September 21, 2010
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
5. For which integers n does Zn have zero div
Math 667
Homework Questions #1
Due: September 7, 2010
Remark: Each answer will be graded on the basis of its correctness, its logical structure, and how
well it is written (style, form, spelling and grammar).
1. Write a complete answer to discussion quest
Math 667
Discussion Questions #10
November 23, 2010
66. Prove that any two groups of order 2 are isomorphic.
67. Prove that isomorphic groups have isomorphic centers.
68. Let G be a group, and let g G. Show that g : G G given by g (x) = gxg 1 is an
automo
Math 667
Discussion Questions #1
August 31, 2010
1. What is a number? What are the essential properties it should possess?
2. Let f (x) = 1/x and g (x) = 1 x.
(a) How many functions can be obtained by composing f and g arbirarily often?
(b) List all such
Math 667
Discussion Questions #3
September 21, 2010
16. Can the division algorithm (p. 18) be applied to a general ring? Why or why not?
17. Should the division algorithm be properly called an algorithm? What is an algorithm?
18. Let b > 0 be an integer.
Math 667
Discussion Questions #4
September 28, 2010
24. Let R be a ring so that every nonzero element of R is also a unit. Prove that every element of
R has a multiplicative inverse. Such a ring is called a division ring.
25. A division ring where the mul
Math 667
Discussion Questions #5
October , 2010
29. Consider the polynomial p = X + 1 in Z2 [X ]. What is p + p? What is p p?
30. Find all of the roots of X 6 1 in Z7 . Factor the polynomial.
31. Find two non-zero polynomials p and q in Z6 [X ] so that de
Math 667
Discussion Questions #6
October 19, 2010
35. Find all of the subgroups of the symmetry group of the equilateral triangle.
36. Find all of the elements of S4 .
37. Find all of the elements of S4 of order 2, that is nd all of the elements S4 with 2
Math 667
Discussion Questions #7
October 26, 2010
42. Find a cyclic subgroup of S3 and determine all of its generators.
43. Is the group of symmetries of the rhombus cyclic?
44. The text describes how the the permutation (12)(34) of a tetrahedron can be f
Math 667
Discussion Questions #8
November 9, 2010
51. Write out the elements of D3 as 2 2 matrices; use the coordinate system from the previous
problem.
52. Let R, and let
M =
cos
sin
sin
.
cos
For (x, y ) R2 , consider the transformation of the plan
Math 667
Discussion Questions #9
November 16, 2010
55. Show that if G is a group and a G, then (a1 )1 = a.
56. Let a and x be elements of the group G. Show that if xa = x, then a = 1.
57. If a is an element of a group G, show that the order of a is the sa
MATH 667
Algebra of Symmetries
Class Policies
Dr. Mike OLeary
Fall 2010
Ofce #1: YR 367
Ofce #2: Terrace Dale, Suite 260
Ofce Phone: 410-704-4757
Email:moleary@towson.edu
Class: Tu 5:00-7:40
Room: Towson High School, Room 111
Section: 101
Ofce Hours: By a