# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

2 Pages

### MIDTERM 1 SOLUTION

Course: MATH 150A, Spring 2003
School: UC Davis
Rating:

Word Count: 457

#### Document Preview

(10/17/07) Math Kuperberg 150a: Modern Algebra First Midterm Solutions 1. Show that every finite group G has an even number of elements of order 3. Solution: In general, for any element g of any group, the orders of g and g-1 are the same. This comes from the fact that gn = e g-n = e. Now, if g G has order 3, then g-1 also has order 3, and we can begin to count off such elements (of order 3) in pairs. There...

Register Now

#### Unformatted Document Excerpt

Coursehero >> California >> UC Davis >> MATH 150A

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
(10/17/07) Math Kuperberg 150a: Modern Algebra First Midterm Solutions 1. Show that every finite group G has an even number of elements of order 3. Solution: In general, for any element g of any group, the orders of g and g-1 are the same. This comes from the fact that gn = e g-n = e. Now, if g G has order 3, then g-1 also has order 3, and we can begin to count off such elements (of order 3) in pairs. There won't be any double counting, since g = g-1 would imply that g has order 2. Since G is a finite group, we are also guaranteed to be finished counting at some point. Thus, we will have an even number of elements of order 3. 2. Consider a product a = xyzw of four transpositions in some symmetric group Sn . Can this product be a 4-cycle? No. A k-cycle must have signature (-1)k-1 . So an even permutation, such as a, cannot be a cycle of even length. Can it be a 5-cycle? Yes. Here's an example: (1 5)(1 4)(1 3)(1 2) = (1 2 3 4 5). Can it be a 9-cycle? No. Each transposition affects two numbers, while every other number is sent to itself. So the product of four transpositions can, at most, act nontrivially on 8 numbers. But a 9-cycle acts nontrivially on 9 numbers. 3. Is the dihedral group D4 isomorphic to the group C4 C2 ? (long): Solution No. D4 C4 C2 . Although the order of both groups is 8 (|D4 | = |C4 C2 | = 8), and both groups have elements of orders 2 and 4; D4 only has two elements of order 4, while C4 C2 has 4. If D4 = {a, a2 , a3 , 1, b, ba, ba2 , ba3 }, only a and a3 are of order 4. On the other hand, C4 C2 = {(1, 1), (x, 1), (x2 , 1)(x3 , 1), (1, y), (x, y), (x2 , y), (x3 , y)} (x, 1), (x3 , 1), (x, y), (x3 , y) (which have order 4). Solution (short): C4 C2 is abelian, but D4 is not. This is a complete solution, but we may also verify these facts: g, h C4 C2 , i, j, k, l N such that g = (xi , y j ) and h = (xk , yl ). Then gh = (xi , y j )(xk , yl ) = (xi+k , y j+k ) = (xk , yl )(xi , y j ) = hg. But in D4 , ab = ba-1 = ba. 4. In the dihedral group D6 (drawn on the board), let f be one of the reflections. Find an element which is conjugate to f (other than f itself) and find an element which is not conjugate to f (also other than f itself, and not equal to 1). Solution: Let r be the rotation of the plane by /3. Then r D6 , and r has order 6. The reflection f has order 2, so it cannot be conjugate to r. On the other hand, conjugating by r (or by a power of r) gives another reflection (through a different line of symmetry).
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

UC Davis - MATH - 150A
Kuperberg (11/14/07)Math 150a: Modern Algebra Second Midterm SolutionsI decided to post students' solutions that I liked for these questions. This way you can see real examples of good work. 1. In the additive group , what is 10 + 25 in set arithm
UC Davis - MATH - 150A
Math 150a: Modern Algebra Solutions to the Final1. If G is a group with a subset A, then conditions for A to be a subgroup are: (1) A is closed under multiplication, (2) A contains the identity 1, and (3) A is closed under taking inverses. Suppose t
NSUOK - ECON - 2213
Chapter 4 US Economy: Private and Public SectorsI.II.IV.Households as Income Receivers A. Functional distribution of 1. Wages and salaries are 72 percent of the total. 2. Proprietors' income (income to self-employed business owners, doctors,
NSUOK - ECON - 2213
Chapter 5 The US in the Global EconomyI.II.Linkages A. Several economic flows link the U.S. economy with the economies of other nations. B. These linkages are 1. Goods and services flows 2. Capital and labor (resource) flows 3. Information and
NSUOK - ECON - 2213
Exam 1 Ch -2 Sample Spring 20081. For economists, the word &quot;utility&quot; means: A) versatility and flexibility. B) rationality. C) pleasure and satisfaction. D) purposefulness. 2. When entering a building, Sam diverts his path to go through an open door
NSUOK - ECON - 2213
NORTHEASTERN STATE UNIVERSITYCollege of Business and Technology Department of Economics Tahlequah-Oklahoma INSTRUCTOR: Ray Murphy Office: Shawnee Theater OCRD -124 W. Shawnee, Suite COffice Hours: 10:00-12:30 MW and 3:00-5:00 M; 10:00-12:30 F Tele
NSUOK - ECON - 2213
CH 3 Questions Class1. State the law of demand and explain why is the other-things-equal assumption critical to it? 2. Give three explanations for the law of demand: 3. Distinguish between the meaning of the expressions &quot;change in demand&quot; and &quot;chan
NSUOK - ECON - 2213
Chapter 4 HandoutElasticityI. Introduction and Overview In chapter 3 we analyzed demand and supply. The demand curve shows an inverse relationship between price of a good and quantity demanded. How does the shape of a demand curve differ between
NSUOK - ECON - 2213
CH 4 Questions Class1. Under what conditions will an increase in price of a product lead to a reduction in total spending for that product.2. The president of a toy company asks you for advice about whether the company should cut the price of its
NSUOK - ECON - 2213
Chapter 1 Limits, Alternatives and ChoicesChapter 1 Limits, Alternatives and ChoicesI. Definition of Economics A. The social science concerned with the efficient use of limited or scarce resources to achieve maximum satisfaction of human materials
NSUOK - ECON - 2213
Chapter 2 The Market SystemEconomic systems differ in two important ways: Who owns the factors of production and the method used to coordinate economic activity. A. The market system our system: 1. There is private ownership of resources. 2. Marke
UC Davis - MATH - 150A
Math 150a: Modern Algebra Homework 10This problem set is due Friday, December 7. Do problems 5.6.3, 5.7.3 (using the counting formula and the stabilizer of a face), 5.7.5, 6.1.6, 6.2.4, 6.2.7, 6.3.2 (left multiplication only), 6.4.2, 6.4.5, and 6.4.
NSUOK - ECON - 2213
Chapter 3 Demand, Supply and Market EquilibriumI.II.Markets Defined A market is an institution or mechanism that brings together buyers (demanders) and sellers (suppliers) of particular goods and services. 1. A market may be local, national, or
NSUOK - ECON - 2213
Chapter 3Supply and Demand: An IntroductionI. Introduction/Overview A. New York City`s food market is much more efficient than its rental housing market. B. Basic economic questions must be answered and can be answered through central planning or m
Cornell - CEE - 3040
Cornell - CEE - 3040
CEE 304 - Uncertainty Analysis in EngineeringSecond ExaminationNovember 12, 2004 You may use text, your notes and calculators. There are 50 points in total, one per minute. 1. (6 pts) True or false, and short answers. a. The sample average is alway
Cornell - CEE - 3040
Cornell - CEE - 3040
Cornell - ILROB - 1220
Motivation Part IISeptember 12, 2007Previous ClassFocused on the person. People are motivated to fulfill certain needs. Everyone has the same set of needs.(E.g. Maslow's Hierarchy, ERG Theory) For certain people, some needs are stronger than
Cornell - ILROB - 1220
Motivation Part 1September 10, 2007DefinitionMotivation: The set of processes that arouse, direct and maintain human behavior toward a goal.Two Types of &quot;Person&quot; Centered Motivation TheoriesContent theories: Offer ways to profile or analyze in
Cornell - ILROB - 1220
The Limitations of PersonalitySeptember 5, 2007Personality Traits Give Useful Information Use the Big 5 (OCEAN) to categorize people ondimensions that are predictive of behavior. Certain traits predict work outcomes Openness to experience
Cornell - ILROB - 1220
Personality in OrganizationsSeptember 3, 2007Person-Situation Interaction FrameworkThis week we will focus on the &quot;Person&quot; side of the equation. Key questionsWhat is personality? How is personality measured? What does personality predict?
Cornell - ILROB - 1220
Person X Situation Interaction FrameworkAugust 29, 2007Behavior at work is complexGoal of Organizational BehaviorTo provide a theoretical framework that can be used to explain, predict and control behavior. Introduce the logic of the Person
Cornell - ILROB - 1220
WELCOMEILROB 122: Introduction to Organizational BehaviorWhat are organizations and why are we here?Answer: Organizations are everywhere and we operate within organizations almost all of the time. Problem: It is difficult to study and be in
Cornell - CEE - 3040
HOMEWORK #10 Selected Solutions P1 1 - = P[Accept Ho | Ho true] fabricator claims beams are premium, and found to be true, both engineer and fabricator are happy = P[Reject Ho | Ho true] Engineer falsely accuses fabricator of selling nonpremium be
Cornell - CEE - 3040
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERINGHomework #8 Due: Monday, October 22, 2007. Read: Sections 6.1 and 6.2, and prob. 34 on p. 252 Devore6 [p. 279 Devore6] for definition of MSE. For confidence intervals please read 7.1-7.2. Goal: We are now
Cornell - CEE - 3040
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING Second Prelim in class on FRIDAY Nov. 9, 2007 open book, open notes. Topics on exam:Continuous random variables including LN/Gumbel/Weibell ( 4.1-4.5), Multivariate Random Variables ( 5.1-5.2, 5.3-5.5),
Cornell - CEE - 3040
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERINGSecond Prelim in class on FRIDAY Nov. 9, 2007. Open book and open notes. Homework #10 Due: Monday, November 5, 2007 Read: Devore 8.1 - 8.2, 8.4 - 8.5; ActivStats chapter 10. Goal: This homework focuses o
Cornell - CEE - 3040
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERINGHomework #11 Due: Nov. 19, 2007. Monday one week after exam. Most problems (particularly P1-P3) are related to material on 2nd exam, so it makes sense to do them THIS week. If you use ActivStats, it is re
Cornell - CEE - 3040
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING letHomework #12 Due: Monday Nov. 26, 2007, in class.Read: 12.1-12.5, and 13.1-13.2, 13.4 (through Devore7 p. 540 [Devore6 p. 600].)This assignment is worth 200 points rather than regular 100 because
Cornell - CEE - 3040
precip 0.77 1.20 3.00 1.62 2.81 2.48 1.74 0.47 3.09 1.31 1.87 0.96 0.81 1.43 1.51 0.32 1.18 1.89 1.20 3.37 2.10 0.59 1.35 0.90 1.95 2.20 0.52 0.81 4.75 2.05
Cornell - CEE - 3040
I.:. - .-.--- - - - - - - ;--3.-&amp;l~uuu\$II;13t&amp; --. :2 :.-...'. . . .-. .I . ...v..3.-.&quot; ,:. . .I.'.'S',.I.. . . &quot; '...:.'., .a5'..&gt;,'_j.&quot;.:.r&quot; '.ma:&quot;.&quot;.I ' :.i).;Z':&quot;: ;%q, . .
Cornell - CEE - 3040
Anna Lee &amp; Jery Stedinger Fall 2006CEE 304 Section 12 Handout SolutionsNonparametric Statistical Tests Example: Q1: Wind speed The appropriate hypotheses are: Ho: d = 0 Ha: d 0 = 5% n = 11, d = -0.67 , sd = 1.12 (a) Wilcoxon Sign Rank Test: S+
Cornell - CEE - 3040
Something to think about:A VIEW FROM ACADEMIADean William Streett College of Engineering Cornell University, 1990. An undergraduate education should provide the following:1. 2. A firm foundation in math and science Beginning of a working knowledg
Cornell - CEE - 3040
Cornell - CEE - 3040
CEE 304 Section 4 Problems (9/20/2006) 1. Transforming Variables Example:f X ( x) = e- xfor x &gt; 0, and 0 otherwiseDevelop the density function for Y = X Y = g(X) = X =&gt; X = Y2, dx = 2y dyOne-to-one transformation: we have a one-to-one relat
Cornell - CEE - 3040
Section #2 solutions Counting Problems:1. A party has 50 persons. 45 of them are boys. 5 are girls. If we select randomly 6 people, a.) What is the probability to have exactly 1 girl? Pr[Select exactly 1 girl] =45 5 50 6 5 10.384b.) To have at
Cornell - CEE - 3040
CEE 304 Section 3 Solutions 1. Suppose the concentrations of a reagent in samples generated in a laboratory are normally distributed. You are told that the reagent concentration must be within 3.0 0.01 mg/l. No concentration outside this range is ac
Cornell - CEE - 3040
CEE 304 Section 4 Solutions 1. Transforming Variables Example:f X ( x) exfor x &gt; 0, and 0 otherwiseDevelop the density function for Y = X Ans. Y = g(X) = X We have a one-to-one relationship between X and Y, so we can use the following formul
Cornell - CEE - 3040
Andrea Gruber &amp; Jery Stedinger Fall 2007Supplemental exercises (p. 94)# 106abcd (Devore 6th edition - #104abcd).15 .67 &gt; .5 =&gt; classify as granite .15 .075 P(B | R1 &lt; R2 &lt; R3) = .33 &lt; 0.5 .0625 b. P(G | R1 &lt; R3 &lt; R2) = .2941 &lt; .5 =&gt; classify as
Cornell - CEE - 3040
CEE 304 Section 5, `07Review of Poisson Processes A Poisson process satisfies three conditions: 1. The probability of an arrival in a short interval t equals t. For small t, the probability of 2 arrivals within t can be neglected. Here is the arriv
Cornell - CEE - 3040
#10 abcf(x) = f(y) = 1/(6-5) = 1a) f(x, y) =1 05x6,5Y6y6otherwisesince f(x) = f(y) = 1/(6-5) = 1, for 5 X 5.75) P(5.25 Yx6, 5y6b) P(5.25 X 5.75, 5.25 independence) c)5.75) = P(5.255.75) = (.5)(.5) = .25 (byIy x 1
Cornell - CEE - 3040
CEE 304 Section 5 (9-27-06)Review of Poisson Processes A Poisson process satisfies three conditions: 1. The probability of an arrival in a short interval t equals t. For small t, the probability of 2 arrivals within t can be neglected. Here is the
Cornell - CEE - 3040
Cornell - CEE - 3040
Andrea Gruber &amp; Jery Stedinger Fall 2007HOMEWORK #6 SOLUTIONS: 1. a) (k+1) =0ske-sds =0sk(-e-s)'ds = sk(-e-s)00(sk)'(-e-s)ds=0k(sk-1) (e-s)ds = k (k)1b) f X xx1ex/,x0E[X] =0x f(x)dx =01xxxxedx=
Cornell - CEE - 3040
CEE 304 Section Example Problems (10-11-06) 1. The maximum annual 24-hour rainfall in Vermont has a mean of 2.5 inches and a standard deviation of 0.7 inches. The distribution is often described by a Gumbel distribution (the appropriate asymptotic d
Cornell - CEE - 3040
Andrea Gruber &amp; Jery Stedinger Fall 2007 HW #12 Selected Solutions # 18abcd Section 12.2, Devore 7 p 466, (Devore 6 p 518) x = temperature (F), y = deflection adjustment factor (y&gt;=0) n =15, xi 1425 , yi 10.68 , xi2 139037.25, yi27.8518 ,987.645
Cornell - CEE - 3040
Andrea Gruber &amp; Jery Stedinger Fall ,07 HOMEWORK # 13 SOULTIONS # 33 Supplemental 15, Devore 7, p 623 (Devore 6 p 694) Observations: 1.7 3.3 5.1 6.9 12.6 14.4 16.4 24.6 26.0 26.5 32.1 37.4 40.1 40.5 41.5 72.4 80.1 86.4 87.5 100.2 Test H 0 : 25.0 ver
Cornell - CEE - 3040
CEE 304 Section #11 Handout (11-17-04)Hypothesis Testing Examples: Two-Sample Test vs. Paired Data Test 1. Sample Final Exam Question: Professor Stedinger's TQM project team decided to investigate the claim that professor X really assigned more hom
Cornell - CEE - 3040
#1 Given: E[ln(S)] = ln(40 ksi); Var[ln(S)] = (0.1)2 E[A] = 1.0 in2; Var[A] = (0.05 in2)2 a) Assuming S and A are independent, find the distirubiton of the yield force: F = S * A F is lognormal because it is the product of two lognormal random variab
Cornell - CEE - 3040
1) {X1,., Xn} can be modeled by a Gamma distribution f X x1x1expx/with a fixed value of the shape parameter .Thus you only need to calculate the value of . [In environmental applications when n is small, may be computed as an average from
Cornell - CEE - 3040
Cornell - CEE - 3040
CEE 304 Uncertainty Analysis in EngineeringPrelim #1 October 5, 2005 You may use the text, your notes, and calculators. There are 50 points in total. You have 50 minutes. 1. (6 pts) Consider three events denoted , , and , where and are independen
Cornell - CEE - 3040
CEE 304 Uncertainty Analysis in EngineeringPrelim 1- October 4, 2006SHOW WORK. You may use the text, your notes, and calculators. There are 50 points in total. You have 50 minutes.1. (8 pts) There are four events of interest, A, B, C and D. You
Cornell - CEE - 3040
CEE 304 - Uncertainty Analysis in EngineeringSecond ExaminationNovember 10, 2006r You may use text, your notes and calculators. No laptops or cell phones. There are 50 points in total, one per minute. 1. (8 pts) A metal chain is constructed from in
Cornell - CEE - 3040
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING2004 Final Examination and Solution 12:00 - 2:30, Thursday, December 9, 2004rExam is open notes and open-book. It lasts 150 minutes and there are 150 points. SHOW WORK!1. (10 pts 2 pts each.) Short an
Cornell - CEE - 3040
Cornell - CEE - 3040
Cornell - CEE - 3040
Cornell - CEE - 3040
Cornell - CEE - 3040
CEE 304-UNCERTAINTY ANALYSIS IN ENGINEERINGFINAL EXAMWednesday, December 8,1993Exam is open notes and open-book. The exam lasts 150 minutes and there are 150 poin is. SHOW WORK!1. (10 points) An engineer must use the cumulative distribution