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 not true if p is not prime; be careful to explain
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 , s2 , s3 , s4 } be the vertices of the square lab
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 (left multiplication only), 6.4.2, 6.4.5, and 6.4.
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 the additive group , what is 10 + 25 in set arithm
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 group G, then X 2 does not mean the same thing as X X
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, and (3) A is closed under taking inverses. Suppose t
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, +)
(R+ , ).
Theorem: Any two cyclic groups of order n ar
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 is worth the same number of points. You must justify or
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 :< g >< 1 > by f(gk ) = k 1 is
an isomorphism.
Let g G. D
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
$111.00 to $145.00.
Mulase, Fuchs, Li, and
- Approved Spri
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 R2 as M.) (b) Show that the set forms a subgroup.
x
(2)
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 R R dened by f(x) = x3 is an isomorphism. (c) Show
tha
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 associative law comes from G. Every element in S has an inv
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
with entries in R with the special property that
(1, 0
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 inverse.
Alternatively, every permutation can be written
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 2.1.4 and 2.1.6 (see also the note in GK4) in addi
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 in the book. Do problems 2.2.16(a,b), 2.2.17, 2.2
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 2.8 and 2.10 of the book. If G is a group and A
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 cyclic. Let k be the first positive integer such that x
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 , and there are two isomorphisms between them: 1 x,
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 that conjugation preserves order: gn = e xgn x-1 =
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: Carefully examine the stitching of a baseball. The stitche
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 of two rotations of R2 by angles and is a rotat
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 reflections. If x and y are both rotations, then they mu
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 group, the orders of g and g-1 are the same. This come