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 it is not closed. Consider any element m. Then m + m =
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 B = cfw_1,
2? (You only need to write
the answer, no pi
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 create the following table: Order 61 62 63 Simple/Not
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 prime, then G is a simple group. Lemma Z: If | G |= pa wit
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 dierent). # 7.15 : Suppose | G |= pq with p and q prime n
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) ab1
= ba e a e a e . e ab1 = ban b1 = beb1 = bb1 =e (b) P
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: a3 , a6 , a9 , a12 For each ak above, gcd(15, k ) = 3.
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 |= 12; | 2 |= 6; | 3 |= 4; | 4 |= 3; | 5 |= 12; | 6 |= 2;
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. (a) This is not a group, since it is not closed. Conside
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 table for GL2 (Z2 ). Let A = 10 11 01 01 ,B= ,C= ,D= ,
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 that
sup(A) = inf A.
First, assume that the infimum of A