HOMEWORK 8: GRADERS NOTES AND SELECTED SOLUTIONS
Graders Notes: In general when you are trying to show that two groups G and H are not isomorphic, its never enough to pick one particular map and show its not an isomorphism. For example, in showing that th
Math 330: Final exam
Problem 1. Complete the following definitions
a) A set G is a group if
b) A subset H of a group G is a normal subgroup if
c) A map (,0 : G > H is a group homomorphism if
cl) A group G acts 011 a set X if
e) A group G is cyc
MATH 330, Exam 1
NO CALCULATORS. Unless otherwise indicated, SHOW ALL WORK.
1. (10 points)
Complete the following definitions.
(a) The set G is a group if . . .
(b) The group G is an abelian group if it additionally satisfies . . .
(c) The subset H
Math 330: Midterm exam
Problem 1. Assume that G and H are groups, n is a positive integer. Decide if the
statement is true or false
b) IflGl = |H| =nthenGEH.
c) 212 < Z. 1
e) If a = (354)(
Math 330 : Practice midterm exam
Problem 1.Complete the following definitions
(1) A set G is a group if .
(2) A group H is cyclic if .
(3) The order of an element g G is .
(4) A permutation Sn is even if .
Problem 2. Represent Q, the quaternio
Math 330 Mid Term, March 3 2006
Q1 Let G be a non empty set, together with a binary operation , which is to be written as
multiplication (a, b) ab, for a, b G.
a. What three properties of the pair (G, ) must be satised for it to be a group?
b. Prove that
Math 330 Questions
(a) Let be a permutation of order 12 of a nite set X . Suppose that can be written
as a product of disjoint cycles, no two of which have the same length. What are the
possible lengths of the cycles in this product?
(b) Suppose that i
Math 330: Abstract Algebra
Sample Final Exam
Look also at the problems on the midterm and sample midterm
1) Dene the following concepts:
a) H is a normal subgroup of G and K is the factor group G/H .
b) I R is a maximal ideal;
c) a R is irreducible.
PRACTICE PROBLEMS FOR THE MATH 330 FINAL
Here is a list of sample problems that you should be comfortable solving quickly
and accurately. I have not included problems involving material from Chapters 18,
20, 21, 22 since we have not covered that material
HOMEWORK 6:GRADERS NOTES AND SELECTED SOLUTIONS
page 84, no. 55 If a = 24 and b = 10 nd the possibilities for ab.
Note that every element x of a b is also an element of the cyclic groups a , b , x divides a = 24 and b = 10, since the only common divisors
c 1 HOMEWORK 5:GRADERS NOTES AND SELECTED SOLUTIONS
Chapter 4, Page 83, No. 21 Let G be a group and let a be an element of G. (1) a12 = e what can we say about the order of a? (2) am = e what can we say about the order of a? (3) that G = 24 and G is cycli
Solutions For Homework 3
Ch. 2 2. Show that the set cfw_5, 15, 25, 35 is a group under multiplication modulo 40. What is the identity element of this group? Can you see any relationship between this group and U (8)? Answer. We need to follow the denition
Abstract Algebra, Homework 1 Solutions Chapter 0 18. Since each pi is a prime we have that pi > 1. Since pi divides the product p1 p2 pn , dividing q = p1 p2 pn + 1 by any one of the primes pi gives a remainder of 1. This means that pi doesnt divide q for
Solutions to Midterm 1
1. The set cfw_1, 2, 3 is not a group under multiplication modulo 4 because it is not closed. In particular, 2 2 = 4 so 2 4 2 = 0. Therefore, cfw_1, 2, 3 cannot be a group. One could also note that 1 is the identit
Solutions HW 3
If you nd any typos in this, please let Professor Shipley know. 1.10. We know that the composition of two rotations is again a rotation. The composition of a reection and a reection is a rotation, a reection followed by a
Math330 : Exam 1 Solution
Problem 1. Decide if each of the following statements is true or false
(1) Q>0 is a group with respect to multiplication. (Notation: Q>0 = cfw_a 2 Q|a > 0.)
(2) If G is a group and a, b 2 G are elements of finite orde