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 subg
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 divis
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 wil
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.
Math 332, Spring 2010 Quiz 1
1. [5 points] (a) List all possible generators for ^") .
Name: _
(b) List all elements of order ") in ^*! .
2. [5 points] Let 5
" %
# $
$ *
% )
& #
' '
( &
) "
* (
. Dete
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. Pr
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.
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
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 = (
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
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 or
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
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 ), (g
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
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
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 a
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 e
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 tha
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 reec
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
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 mo
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, i
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)
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 nonze
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 )