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by: Vivien Bradtke V


Vivien Bradtke V
Texas A&M
GPA 3.6


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This 3 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 415 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/226059/math-415-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15
Math 415 Review for Examination 1 Fall 2006 Modern Algebra I This course contains many new concepts One way to organize those concepts in your mind is to make yourself lists Here are some suggestions about how to review for the rst examination De nitions Make yourself a list of the main de nitions By consulting the table of contents in the textbook7 you will identify the key concepts of relation7 binary operation7 isomorphism7 group7 subgroup7 cyclic group7 and generating set By looking through each section7 you will nd various other concepts identi ed by the word De nition77 in boldface in the left hand margin There is a list of notations in the back of the book starting on page 487 The list is arranged by rst appearance of the symbol in the book Since we have covered through page 74 of the textbook7 you should recognize all the symbols on page 487 and the rst half of page 488 A de nition is not much use without some examples to illustrate it For each concept7 can you give an example that satis es the de nition Can you give an example that fails to satisfy the de nition The next step in reviewing is to see how the de nitions t together Typically the relations among de nitions are expressed as theorems Theorems Make yourself a list of the main theorems These are identi ed in the textbook by the word Theorem77 in boldface in the left hand margin You will also nd theorems identi ed by the word Corollary typically7 corollaries are important special cases of more general theorems Theorems usually say something like7 Every X is a Y 7 or7 Every X has property Pf7 Do you know examples of all the concepts mentioned in the theorem Do you know a concrete situation to which the theorem applies Can you paraphrase the statement of the theorem What is the signi cance of the theorem For instance7 the rst theorem stated in the book is Theorem 022 on page 7 It says that every equivalence relation on a set induces a partition of the set7 and conversely7 every partition of a set induces an equivalence relation on the set The signi cance ofthe theorem is that it says that two apparently different concepts are essentially the same An example that you should have in mind is equivalence modulo n on the integers the corresponding partition consists of sets of integers that leave the same remainder after division by n The theorem implies that exercises 23727 on page 10 of the textbook could September 247 2006 Page 1 of 3 Dr Boas Math 415 Review for Examination 1 Fall 2006 Modern Algebra I be rephrased instead of nd the number of different partitions of a set the wording could be nd the number of different equivalence relations on a set the second wording sounds like a harder problem but it is really the same problem in disguise To be sure you understand a theorem you should also think about what the theorem does not say For instance Theorem 61 on page 59 says Every cyclic group is abelian77 The theorem does not say anything about non cyclic groups Do there exist non cyclic groups that are abelian Yes the Klein 4 group is a non cyclic abelian group In other words the converse of the theorem is not true After you are sure that you understand the meaning of a theorem and you know illustrative examples you should ask yourself why the theorem is true How does the proof go Can you give a one sentence paraphrase of the main idea in the proof For instance in the theorem mentioned above about equivalence relations the idea is that an equivalence relation determines equivalence classes and the equivalence classes form a partition of the set In the proof of the theorem that every cyclic group is abelian the idea is simply that different powers of the same element always commute with each other Examples Abstract concepts are illuminated by concrete examples The most important notion in the rst part of the course is the notion of a group Making yourself a list of speci c groups that we have encountered is a good way to review In particular we know a complete listiup to isomorphismiof nite groups of small order Groups of order 2 are all isomorphic to Z2 Groups of order 3 are all isomorphic to Z3 We know two non isomorphic groups of order 4 namely Z4 and the Klein 4 group and every group of order 4 is isomorphic to one of these two Preview of coming attractions We will soon learn that all groups of or der 5 are isomorphic to Z5 In particular this means that all groups of order 5 or smaller are abelian We will soon learn that there are two non isomorphic groups of order 6 one is Z6 and the other is the group of symmetries of an equilateral triangle the latter group is non abelian We know a complete list of nite cyclic groups these are isomorphic to Zn for some positive integer n We also know that every in nite cyclic group is isomorphic to Z September 24 2006 Page 2 of 3 Dr Boas Math 415 Review for Examination 1 Fall 2006 Modern Algebra I We know various examples of non cyclic in nite groups such as Q un der addition Q4r under multiplication and invertible matrices under matrix multiplication You will nd other examples by looking through the book and the exercises Exercises You solidify your understanding of mathematics by solving prob lems If there were homework exercises that you had trouble with you should review the concepts involved Solving additional exercises can only help im prove your understanding many ofthe odd numbered exercises have answers in the back of the book so that you can check your solutions About the examination As I mentioned in class the rst examination covers sections 077 in the textbook There will be ve truefalse questions worth ve points each ve ll in the blanks questions worth ve points each and three essay type questions worth fteen points each There will be ve style77 points based on how well written your essays are I expect that most papers will receive the style points if the solutions are clearly explained and well organized September 24 2006 Page 3 of 3 Dr Boas


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