(1) Let = (123)(145). Write
99
Assignment 8 Solutions in disjoint cycle form.
Solution. We first write in disjoint cycle form: = (14523). This is a 5-cycle, so it has order 5. It follows that 100 = . So 99 = -1 = (13254). (2) Let = (1, 3, 5, 7, 9, 8, 6)(
Practice Midterm
All questions are worth ten points. In addition to the questions, there will
be an additional ten points to be awarded for style and clarity in writing.
1) Denitions
1. Dene the index |G : H| of a subgroup H < G.
2. Dene the external dire
Lecture 7 Notes
How to learn Abstract Algebra
Here are some remarks about how to do algebra problems.
1. When you are asked to prove a statement you must not assume that the
statement is true.
2. Never assume a group is Abelian. Some people begin their ar
Lecture 4 Notes
How to learn Proofs
HOW TO GET STARTED
Begin a proof by rewriting what you are given and what you are asked to prove
in a more convenient form. Often this involves converting word to symbols and
utilizing the definitions of the terms used
Lecture 4 Notes
How to learn Proofs
HOW TO GET STARTED
Begin a proof by rewriting what you are given and what you are asked to prove
in a more convenient form. Often this involves converting word to symbols and
utilizing the definitions of the terms used
Lecture 1 Notes
Reasons to take this class
1. Even though many students take a course is discrete math where they study
various proof techniques many of them seem not to absorb this material well.
Abstract algebra provides them much more practice at this
Lecture 1 Notes
Reasons to take this class
1. Even though many students take a course is discrete math where they study
various proof techniques many of them seem not to absorb this material well.
Abstract algebra provides them much more practice at this
Final Exam Prep
All questions are worth ten points, unless otherwise indicated. In addition to
the questions, there will be an additional ten points to be awarded for style
and clarity in writing. The problems that appear on the actual exam may be
slightl
Final Exam Prep
All questions are worth ten points, unless otherwise indicated. In addition to
the questions, there will be an additional ten points to be awarded for style
and clarity in writing. The problems that appear on the actual exam may be
slightl
Lecture 8 Notes Any permutation may also be written or "decomposed" as a product of two-cycles. This product would usually not be disjoint, and it need not be unique. However, if a permutation can be written as an odd number of two-cycles, any other decom
Lecture 6 Notes A Cyclic Group is a group which can be generated by one of its elements. That is, for some a in G, G=cfw_an | n is an element of Z Or, in addition notation, G=cfw_na |n is an element of Z This element a (which need not be unique) is called
Lecture 5 Notes Abelian Groups are groups which have the Commutative property, a*b=b*a for all a and b in G. This is so familiar from ordinary arithmetic on Real numbers, that students who are new to Abstract Algebra must be careful not to assume that it
Lecture 3 Notes Groups may be Finite or Infinite; that is, they may contain a finite number of elements, or an infinite number of elements. Also, groups may be Commutative or Non-Commutative, that is, the commutative property may or may not apply to all e
Assignment 15 Solutions (1) Let G = R[x] = cfw_an x + + a2 x + a1 x + a0 | ai R be the group (under addition) of polynomials with coefficients in R. Let f denote the antiderivative of f with integration constant 0. (a) Show that (f ) = f defines a homomor
Assignment 14 Solutions (1) Prove the external direct product of a finite number of groups is a group. Proof. Let (G1 , 1 ), (G2 , 2 ) . . . , (Gn , n ) be groups. The external direct product is the set G1 Gn = cfw_(g1 , g2 . . . , gn )|gi Gi . We define
Assignment 13 Solutions (1) Let H = cfw_(1), (12)(34), (13)(24), (14)(23). How many left cosets of H are there in S4 ? Solution. By Lagrange's Theorem, the number of distinct left cosets of H in S4 is (2) Suppose a is an element of a group and |a| = 15. F
Assignment 11 Solutions (1) Prove that a permutation is even if and only if its inverse is even. Conclude that a permutation is odd if and only if its inverse is odd. Proof. Let be any permutation. Let 1 , 2 , . . . , k be transpositions such that = 1 2 k
Assignment 9 Solutions (1) Let m be a positive integer and let = (a1 , a2 , . . . , am ). Prove that for all 1 i m i (aj ) = ai+j am
mod m
if i + j mod m = 0 if i + j mod m = 0
()
Proof. First suppose i = 1. In this case () reads (aj ) = aj+1 mod m am if
Algebra:
Chapter 0
Paolo Aluffi
Graduate Studies
in Mathematics
Volume 104
American Mathematical Society
Algebra:
Chapter 0
Algebra:
Chapter 0
Paolo Aluffi
Graduate Studies
in Mathematics
Volume 104
American Mathematical Society
Providence, Rhode Island
E