MA 541 Modern Algebra I
Solutions to HW #1 1. Which of the following are groups under the given operation? Justify your answers. (a) The collection of odd integers under +. This is not a group, since
Assignment 2, Math 20300
(1) Draw a picture for the following sets in the xy plane:
(a) cfw_0 cfw_1,
(b) (a, b) R,
(c) R cfw_1,
(d) cfw_1 [1, 1],
(e) What is the cartesian product of A = cfw_0, 1 and
Math 541
Solutions to HW #10 1. Prove that if G is a simple group such that 61 | G | 70, then G is a cyclic group. [Feel free to use the Sylow theorems.] Referencing the lemmas from homework 9, we can
Math 541
Solution to HW #9 Assignment: Prove that if G is a simple group of size less than 60, then G is cyclic of prime size. Recall from class the following lemmas: Lemma P: If | G |= p with p a pri
Math 541
Solutions to HW #8 The following are from Gallian, Chapters 7 and 9. The numbering is rst given for the 7th edition, and is then followed by the 6th editions numbering (if the numbering is di
Math 541
Solutions to HW #7 1. Let G be a group and let a be an element of G with order n. (a) Prove (bab1 )n = e. (bab1 )n = bab1 bab1 bab1 . bab1 = ba (b
1
b) a ( b
n times 1
b) a (b1 b) . (b1 b) ab
Math 541
Solutions to HW #6 The following are from Gallian, Chapters 4 and 5 (6th edition). # 4.8 : Let a be an element of a group and let | a |= 15. Compute the orders of the following elements of G:
Math 541
Solutions to HW #5 The following are from Gallian, Chapter 3 (6th edition). In this solution set, |a| denotes the order of the element a (i.e. ord(a). # 1: Z12 : | Z12 |= 12; | 0 |= 1; | 1 |=
Math 541
Solutions to HW #4 1. Use the Euclidean algorithm to compute the greatest common divisor of 320353 and 257642.
320353 = 257642(1) + 62711 257642 = 62711(4) + 6798 62711 = 6798(9) + 1529 6798
Math 541
Solutions to HW #3 1. Gallian Chapter 2: 3,5 #3 Show that (a) cfw_1, 2, 3 under multiplication modulo 4 is not a group, but that (b) cfw_1, 2, 3, 4 under multiplication modulo 5 is a group. (
Math 541
Solutions to HW #2 1. Let GL2 (Z2 ) denote the collection of 2 2 matrices with entries in Z2 which have non-zero determinant.(We listed these matrices out in class.) (a) Make a multiplication
Math 203 Problem Set 1 Solutions
Monday, October 10, 2005
Problem 1
Statement
Let A be a nonempty subset of R. Define A = cfw_x ; x A. Show that
sup(A) = inf A
inf(A) = sup A.
Solution
We shall show t