Math 150a: Modern Algebra Homework 6 Solutions
GK1. Show that if G is a finite group and p is a prime number, then the number of elements of order p in G is divisible by p - 1. The result is certainly
Math 150a: Modern Algebra Homework 10 Solutions
5.6.3 (a) Exhibit the bijective map (5.6.4) explicitly, when G is the dihedral group D4 and S is the set of vertices of a square. Solution: Let S = {s1
Math 150a: Modern Algebra Homework 10
This 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
Kuperberg (11/14/07)
Math 150a: Modern Algebra Second Midterm Solutions
I decided to post students' solutions that I liked for these questions. This way you can see real examples of good work. 1. In
Math 150a: Modern Algebra Homework 7
This problem set is due Wednesday, November 14. Do problem 5.9.2 and the following problems: GK1. A review problem in set arithmetic. a. If X is a subset of a grou
Math 150a: Modern Algebra Solutions to the Final
1. 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, an
MAT 150A HW2 Solutions, Fall 2014
(1) (a) Let e1 =
and e2 = (0, 1)T be the standard basis vectors of R2 . Let A
GL2 (R) be a matrix which xes the x-axis. In other words, A is an invertible 2 2 matrix
Math 150A, Lecture 7
An isomorphism f : G1 G2 is a bijection such that f(x G1 y) = f(x) G2 f(y).
Ex. f :< i >< (1234) > dened by f(ik ) = (1234)k
Example: Let g be an element of G and |g| = . f :<
Math 150A, Lecture 8
The inverse of an isomorphism is an isomorphism.
Def. Two groups are isomorphic if there is an isomorphism between them. If G is
isomorphic to H, we write G H.
Examples: (R, +)
Math 132a: Stochastic Processes
Final Exam
(3/22/03)
Write your name and student ID number in the upper right-hand corner of this sheet and write your initials
on each page of your exam.
Each problem
Math 150A, Lecture 4
The determinant of the permutation matrix is called the sign of the permutation. If it
is +1, then it is even; if it is 1, then it is odd.
Every element of order 2 is its own in
HOMEWORK 2, DUE 10/20/2014, MONDAY 10AM, MATH 150A
Suggested Readings: Sections 1.5, 2.1-2.6
(1) (a) Determine the subset of GL2 (R) that xes the x-axis. (Notice that each element
M GL2 (R) acts on x
HOMEWORK 3, DUE 10/27/2014, MONDAY 10AM, MATH 150A
Suggested Readings: Sections 2.5, 2.6.
(1) (a) Show (but not turn in) (R , ), where R is the set of non-zero real numbers is a
group. (b) Show that f
Math 150A, Lecture 5
Subgroup: Let S be a subset of a group G. S is a subgroup (i) it is closed under the
group operation, (ii) for every a, b S, we have ab1 S. (In particular, S contains
e. The asso
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT 150A, Modern Algebra
Algebra by Michael Artin; Addison Wesley; 2nd
edition (August 13, 2010); ISBN-10: 0132413779;
ISBN-13: 978-0132413770; Price ranges from
$11
Math 150a: Modern Algebra Homework 1
This problem set would ideally have been due Wednesday, October 3. But since I am running late, it can be turned with no penalty on Friday, October 5. Do problems
Math 150a: Modern Algebra Homework 2
This problem set is due Wednesday, October 10. Starred problems may be harder and will be counted as extra credit. This includes both my starred problems and those
Math 150a: Modern Algebra Cross-sections and complements
Since the book does not much discuss cross-sections and complements, here are some notes that may be helpful. This topic is related to sections
Math 150a: Modern Algebra Homework 3
2.2.13: Prove that every subgroup of a cyclic group is cyclic. Solution: Let H be a subgroup of the cyclic group Cn =< x |xn = 1 >. We want to show that H is cycli
Math 150a: Modern Algebra Homework 4 Solutions
2.3.16: Give an example of two isomorphic groups such that there is more than one isomorphism between them. Solution: Well, at first thought, Z/3 C3 , a
Math 150a: Modern Algebra Homework 5 Solutions
2.4.19: Prove that if a group contains exactly one element of order 2, then that element is in the center of the group. Solution: The key fact here is th
Math 150a: Modern Algebra Homework 7 Solutions
5.9.2 Identify the group of symmetries of a baseball, taking the stitching into account and allowing orientation reversing symmetries. Solution: Carefull
Math 150a: Modern Algebra Homework 8 Solutions
4.5.4 (b) Is O(2) isomorphic to the product group SO(2) {I}? Is O(3) isomorphic to SO(3) {I}? Solution: The group SO(2) is abelian [because the product
Math 150a: Modern Algebra Homework 9 Solutions
GK1. For which n is the dihedral group Dn equiconjugate with O(2)? Solution: Let x, y Dn be conjugate in O(2). Then both x and y are rotations or reflec
Kuperberg (10/17/07)
Math 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 grou
Math 150a: Modern Algebra Homework 5
This problem set is due Wednesday, October 31. Do problems 2.4.19, 2.5.3 (equivalently, the number of partitions), and 2.6.10(a), in addition to the following: GK1
Math 150A, Lecture 6
Every cyclic group is abelian.
The Klein 4-group is abelian but NOT cyclic. Every non-identity element in the Klein
4-group has order 2. Every 2 non-identity elements in the Kle