Homework 11 due 11/20/15
Math 403
You should be reading sections 3.3-3.5 of the text by Rotman.
For EXAM 2, you are responsible for the material on Homeworks 5-10, and for material in lectures from October 2 until November 11 (we covered the 1st Isomorphi
MATH 402 Final
(/{5 SCM
A
Tuesday, 20 May 2008
Instructions: Each problem is worth 10 points. The total will then be scaled out of 200 points.
Please do one problem on each sheet.
1.
2.
10.
Prove by mathematical induction that 27132 1 is a multipl
Exam I
April 6, 2010
Joel M. Cohen
Math 402
9:30-10:45
Name:
Read the problems very carefully! Put your nal answer to each problem in a
BOX if the problem is a computation.
Recall that Dn = cfw_rk , rk f, k = 0, 1, . . . , n 1 with the rules: rk f = frk ,
Exam II
March 30, 2006
Joel M. Cohen
Math 403
3:30 - 4:45 p.m.
Name:
Read the problems very carefully.
1. (10 points) Assume that G is a group with subgroups H and K where |H | = 12
and |K | = 35. Find the number of elements in the set HK , and explain wh
Exam I
March 1, 2007
Joel M. Cohen
Math 403
11:00 a.m. - 12:15 p.m.
Name:
Read the problems very carefully. The real numbers are R, the rationals Q. The
dihedral group Dn is cfw_rk , rk f, k = 0, 1, . . . , n 1 with the rules: rk f = frk , f 2 =
rn = e, t
Homework 8 due 10/30/15
Math 403
You should be reading sections 2.7, 3.1 of the text by Rotman.
1. Show there is a group isomorphism R/Z S 1 (give the isomorphism explicitly).
=
Show that under the correspondence theorem, the subgroups n S 1 (the cyclic
1
Homework 10 due 11/13/15
Math 403
You should be reading sections 3.1-3.3 of the text by Rotman.
1. (a) Prove that M2 (Z), the set of 2 2 matrices with integer entries, forms a
ring with 1 when equipped with the usual operations of addition and multiplicat
Homework 7 due 10/23/15
Math 403
You should be reading sections 2.6-2.7 of the text by Rotman.
1. Rotman, #2.104.
2. (a) Using the denition of gcd in section 1.3 of Rotman, prove that the gcd (a, b)
of two non-zero integers a, b is the unique positive num
Homework 6 due 10/16/15
Math 403
You should be reading sections 2.5-2.7 of the text by Rotman.
1. Rotman, #2.72.
2. (a) Let nZ and mZ are two subgroups of Z. Prove that mZ nZ if and only if
n|m.
(b) Using the Correspondence Theorem, describe explicitly al
Homework 3 due 09/25/15
Math 403
You should be reading sections 2.2-2.4 of the text by Rotman.
1. (a) Let G be a nite group. Show that there exists a positive integer n such
that an = e for all a G. (We call the smallest such n the exponent of G.)
(b) Fin
Homework 2 due 09/18/15
Math 403
1. (a) Rotman, #1.32.
(b) Rotman, #1.33.
2. (a) Give the detailed proof by induction that (x + y)n = n n xr y nr for any
r=0 r
integer n 0. Use the identity n1 + n1 = n for 1 r n 1 we proved in
r1
r
r
class.
(b) Rotman, #1
Homework 4 due 10/02/15
Math 403
You should be reading sections 2.3-2.6 of the text by Rotman.
1. (a) Rotman, #2.42.
(b) Find an innite group such that every element has nite order.
2. Rotman, #2.52.
3. Rotman, #2.56 and #2.57.
4. (a) Let H = (12) , a sub
Homework 1 due 09/11/15
Math 403
1. Let A and B be subsets of a set X, with complements A and B . Prove the de
Morgan laws
(a) (A B) = A B
(b) (A B) = A B
(c) (A\B) (B\A) = (A B)\(A B).
For each of these equalities, draw a picture illustrating it.
2. For
1
Solution Manual for
A First Course in Abstract Algebra, with Applications
Third Edition
by Joseph J. Rotman
Exercises for Chapter 1
1.1 True or false with reasons.
(i) There is a largest integer in every nonempty set of negative integers.
Solution. True
Final Exam
May 16, 2009
Joel M. Cohen
Math 403
8:00 - 10:00 a.m.
Name:
Read the problems very carefully. The real numbers are R, the rationals Q, and
the complex numbers C.
1. (25 points) (a) Let N, H be the two cyclic subgroups of S3 generate by (1 2 3)