Homology and cohomology Prof. Kathryn Hess Series 10
Spring 2009
The purpose of the first part of this series is to show that the integral homology groups of a space X determine the homology and cohomology groups of X with arbitary coefficients. The state
Instructions
(1) This question paper consists of two parts: Part A and Part B and carries a total of 100 Marks. (2) There is no negative marking. (3) Part A carries 20 multiple choice questions of 2 marks each. Answer all questions in Part A. (4) Answers
27. Fields from quotient rings In Lecture 26 we have shown that the quotient ring R[x]/(x2 + 1)R[x] is isomorphic to C, so, in particular, it is a field, while R[x]/(x2 - 1)R[x] is not a field. The reason we did not get a field in the second case is clear
26. Examples of quotient rings In this lecture we will consider some interesting examples of quotient rings. First we will recall the definition of a quotient ring and also define homomorphisms and isomorphisms of rings. Definition. Let R be a commutative
25. Ideals and quotient rings We continue our study of rings by making analogies with groups. The next concept we introduce is that of an ideal of a ring. Ideals are ring-theoretic counterparts of normal subgroups. Recall that one of the main reasons why
25. Ideals and quotient rings We continue our study of rings by making analogies with groups. The next concept we introduce is that of an ideal of a ring. Ideals are ring-theoretic counterparts of normal subgroups. Recall that one of the main reasons why
24. Rings 24.1. Definitions and basic examples. Definition. A ring R is a set with two binary operations + (addition) and (multiplication) satisfying the following axioms: (A0) (A1) (A2) (A3) (A4) (M0) (M1) (D1) (D2) R is closed under addition addition is
23. Quotient groups II 23.1. Proof of the fundamental theorem of homomorphisms (FTH). We start by recalling the statement of FTH introduced last time. Theorem (FTH). Let G, H be groups and : G H a homomorphism. Then G/Ker (G). ( ) = Proof. Let K = Ker and
22. Quotient groups I 22.1. Definition of quotient groups. Let G be a group and H a subgroup of G. Denote by G/H the set of distinct (left) cosets with respect to H. In other words, we list all the cosets of the form gH (with g G) without repetitions and
22. Quotient groups I 22.1. Definition of quotient groups. Let G be a group and H a subgroup of G. Denote by G/H the set of distinct (left) cosets with respect to H. In other words, we list all the cosets of the form gH (with g G) without repetitions and
21. Permutation groups II 21.1. Conjugacy classes. Let G be a group, and consider the following relation on G: given f, h G, we put f h there exists g G s.t. h = gf g -1 . Thus, in the terminolgy from Lecture 20, f h h is a conjugate of f . Definition. Th
21. Permutation groups II 21.1. Conjugacy classes. Let G be a group, and consider the following relation on G: given f, h G, we put f h there exists g G s.t. h = gf g -1 . Thus, in the terminolgy from Lecture 20, f h h is a conjugate of f . Definition. Th
Lecture 19
6
Abstract algebra and coding theory (continued)
1 2
Groups Rings and fields
1 / 14
Groups XXVIII
We sometimes use the following terminology:
Definition
A left coset g = gH in a group G for a subgroup H is also called a class. An element x g is
1.
Even solutions for Homework 7
1
1.
Even solutions for Homework 7
14.4: #2: negative 14.4 #20: 14.4 #24: 14.4 #34:
L f = .3( K ).7 i + .7( K ).3 j L
z = 2xcos(x2 + y 2 )i + 2ycos(x2 + y 2 )j
1 2i
+ 1j 2
14.4 #54: A. 15/ 10, B. 9i + 12j 14.6 #2:
t3 -2 t
Math 103B Homework Solutions HW5
Jacek Nowacki May 13, 2007
Chapter 19
Problem 2. Prove that a nonempty subset U of a vector space V over a field F is a subspace of V if, for every u and u in U and every a in F , u + u U and au U . Proof. Remark: Let's be
Math 103B Homework Solutions HW5
Jacek Nowacki May 13, 2007
Chapter 19
Problem 2. Prove that a nonempty subset U of a vector space V over a field F is a subspace of V if, for every u and u in U and every a in F , u + u U and au U . Proof. Remark: Let's be
Math 103A Homework Solutions HW4
Jacek Nowacki February 27, 2007
Chapter 6
Problem 4. Show that U (8) is not isomorphic to U (10). Proof. First, we note that U (8) = cfw_1, 3, 5, 7 and U (10) = cfw_1, 3, 7, 9. Hence, we can't use orders as they are the sa
Math 103B Homework Solutions HW3
Jacek Nowacki April 30, 2007
Chapter 17
Problem 4. Suppose that f (x) = xn + an-1 xn-1 + . . . + a0 Z[x]. If r is a rational number and x - r divides f (x), show that r is an integer. Proof. x-r divides f (x) implies that
Math103B Homework Solutions HW1
Jacek Nowacki April 26, 2007
Chapter 13
Problem 8. Describe all zero-divisors and units of R = Z Q Z. Proof. All elements of R are triplets of the form (z1 , q, z2 ), where z1 , z2 Z and q Q. Now, we note that both operatio
The Fourier Transform and Equations over Finite Abelian Groups
An introduction to the method of trigonometric sums
LECTURE NOTES BY Lszl Babai a o Department of Computer Science University of Chicago
December 1989 Updated June 2002
VERSION 1.3
The aim of
Finite Abelian Groups
1 Preliminaries
Denition 1. Let H1 , H2, . . ., Hk be subgroups of G. We say that G is the internal direct product of H1 , . . . , Hk if the following hold. 1. Hi G, for all Hi
2. G = H1H2 Hk 3. Hi H1 Hi Hk = cfw_e (H1 Hi Hk means th
CONGRUENCE The existence of solutions to binary quadratic equations depends in general on the existence of an integer u such that a quadratic expression a2u2 + a1u + a0 (with integer coefficients) is divisible by a fixed integer n. We now examine the prob
Chapter 6
Finite Abelian Groups
In this chapter, we shall give a complete classification of finite abelian groups. In so doing, we shall observe how the assumption that our binary operation is commutative brings considerable restriction and so makes abeli
Invent.math. 87, 253-302 (1987)
/n/2e/o~/8s
mathematicae
9 Springer-Verlag1987
SK I
of finite abelian groups, II
Roger C. Alperin 1., R. Keith Dennis 2., R. Oliver 3, and Michael R. Stein 4. t Department of Mathematics, The Universityof Oklahoma, Norman,
Invent.math. 87, 253-302 (1987)
/n/2e/o~/8s
mathematicae
9 Springer-Verlag1987
SK I
of finite abelian groups, II
Roger C. Alperin 1., R. Keith Dennis 2., R. Oliver 3, and Michael R. Stein 4. t Department of Mathematics, The Universityof Oklahoma, Norman,
Mathematics Department
Mark Hamilton Room 7.548 V57 WT 2011/12
Advanced Higher Mathematics for INFOTECH Exercise sheet 10
1. a) Let G be a group such that a2 = e for all elements a G. Prove that G is abelian. b) Let G be a group and H G a subgroup of inde
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