Quiz 1 Practice Problems: Cyclic Groups
Math 332, Spring 2010 These are not to be handed in. The quiz will be on Tuesday. 1. Find all generators of Z6 , Z8 , and Z20 . 2. List all elements of the subgroup 30 in Z80 . 3. If |a| = 60, what is the order of a
Homework 8
Math 332, Spring 2010
A These problems must be written up in L TEX, and are due this Thursday, May 13.
1. Consider the ring M2 (Z2 ) = ab : a, b, c, d Z2 . cd
(a) This ring has 9 zero divisors. List them. (b) The ring has 6 units. List them. (c
Takehome Midterm
Math 332, Spring 2010 This midterm has six problems, each of which has several parts. Your solutions must be A written up in L TEX, and are due on Friday, April 30. Late solutions will not be accepted. Because this is a midterm, you are r
Quiz 1 Practice Problems: Permutations
Math 332, Spring 2010 These are not to be handed in. The quiz will be on Tuesday. 1. Write each of the following permutations as a product of disjoint cycles: a. 12345678 23451786 b. 12345678 . 13876524
2. Compute ea
Math 332, Spring 2010 Quiz 1
1. [5 points] (a) List all possible generators for ^") .
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(b) List all elements of order ") in ^*! .
2. [5 points] Let 5
" %
# $
$ *
% )
& #
' '
( &
) "
* (
. Determine the order of 5.
3. c5 pointsd Write a$ &ba" # &ba
Quiz 2 Practice Problems
Math 332, Spring 2010
Isomorphisms and Automorphisms 1. Let C be the group of complex numbers under the operation of addition, and dene a function : C C by (a + bi) = a bi. Prove that is an automorphism of C. 2. Let G be an abelia
Math 332, Spring 2010 Quiz #
Name: _
1. [8 points] Let Z be the group of integers under addition, and let K be the group under the operation of matrix multiplication. Prove that K is isomorphic to Z.
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+ + Z "
2. [8 points] Consider the following subgro
Quiz 3 Practice Problems
Math 332, Spring 2010
Questions on Groups 1. Let G = U (15), and let N = cfw_1, 4. Determine the isomorphism type of G/N . 2. Let G = S4 , and let N = cfw_ e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) . Determine the order of the element
Quiz 3 Practice Problem Solutions
Math 332, Spring 2010
Questions on Groups 1. The group G/N has four elements: N = cfw_1, 4, 2N = cfw_2, 8, 7N = cfw_7, 13, 11N = cfw_11, 14.
Since (2N )2 = (7N )2 = (11N )2 = N , the isomorphism type is Z2 Z2 (or V ). 2.
Cosets and Lagranges Theorem
Study Guide Outline
1. Cosets Let G be a group, and let H G. Given an element g G, the left coset of H containing g is the set gH = cfw_gh : h H Each left coset of H has the same size as H , and the left cosets of H form a pa
Direct Products
Study Guide Outline
1. External Direct Products If G and H are groups, the direct product of G and H , denoted G H (or G H in the book), is dened as follows: The elements of G H are ordered pairs (g, h), where g G and h H . The operation o
Homework 7
Math 332, Spring 2010
A These problems must be written up in L TEX, and are due this Thursday, May 6.
1. (a) Let Z 2 = cfw_a + b 2 : a, b Z. Prove that Z 2 is a subring of the real numbers. (b) Let M2 (R) be the ring of 2 2 matrices with real
Homework 6 Solutions
Math 332, Spring 2010 Problem 1. (a) Proposition. If G and H are groups, then G H H G. Proof. Dene : G H H G by (g, h) = (h, g ). Clearly is bijective. Moreover, if (g1 , h1 ), (g2 , h2 ) G H , then (g1 , h1 )(g2 , h2 ) = (g1 g2 , h1
Homework 6
Math 332, Spring 2010
A These problems must be written up in L TEX, and are due this Thursday, March 18.
1. (a) If G and H are groups, prove that G H H G. (b) Let G and H be groups, and let A G and B H . Prove that A B G H . (c) Let G, G , H ,
Homework 1
Math 332, Spring 2010
A These problems must be written up in L TEX, and are due next Thursday.
1. Let G be a group. Two elements a, b G are said to be conjugate if there exists an element c G for which b = c1 ac. (a) Prove that are conjugate is
Homework 1 Solutions
Math 332, Spring 2010 Problem 1. Let G be a group. Two elements g, h G are said to be conjugate if there exists an element a G for which h = a1 ga. (a) Proposition. Conjugacy is an equivalence relation on G. Proof. We must prove that
Homework 2
Math 332, Spring 2010
A These problems must be written up in L TEX, and are due next Thursday, February 11.
1. Prove that every nite group with an even number of elements has at least one element of order two. 2. Let G be a group, and let a be
Homework 2 Solutions
Math 332, Spring 2010 Problem 1. Proposition. Every nite group with an even number of elements has at least one element of order two. Proof. Let G be a nite group, and suppose that G does not have any elements of order two. We shall p
Homework 3
Math 332, Spring 2010
A These problems must be written up in L TEX, and are due next Thursday, February 18.
1. Recall that the dihedral group Dn has n rotations e, r, . . . , rn1 and n reections s, rs, . . . , rn1 s, where r and s satisfy the r
Homework 3 Solutions
Math 332, Spring 2010 Problem 1. (a) Proposition. If n is odd, then all the reections in Dn lie in the same conjugacy class. Proof. Every reection in Dn has the form rk s for some power k . Conjugating such a reection by r gives r(rk
Homework 4
Math 332, Spring 2010
A These problems must be written up in L TEX, and are due next Thursday, March 4.
1. For n 3, prove that every element of An can be expressed as a product of one or more 3-cycles. 2. In a perfect rie shue, a deck of cards
Homework 4 Solutions
Math 332, Spring 2010 Problem 1. Proposition. If n 3, then every element of An can be written as a product of one or more 3-cycles. Proof. Let An . Since is an even permutation, it is possible to express as the product of an even numb
Homework 5
Math 332, Spring 2010
A These problems must be written up in L TEX, and are due this Thursday, March 11.
1. Let G be the following subgroup of GL(2, R): G= a b : a, b R and (a, b) = (0, 0) . ba
Prove that G is isomorphic to C# , the group of no
Homework 5 Solutions
Math 332, Spring 2010 Problem 1. Proposition. Let G be the following subgroup of GL(2, R): G= a b : a, b R and (a, b) = (0, 0) . ba
Then G is isomorphic to C# , the group of nonzero complex numbers under multiplication. Proof. Dene :
Isomorphisms and Automorphisms
Study Guide Outline
1. Isomorphisms Let G and H be groups. An isomorphism from G to H is a function : G H satisfying the following conditions: is a bijection, and (xy ) = (x)(y ) for all x, y G. If there exists an isomorphis