Selected Solutions to Math 4107, Set 4
November 9, 2005
Page 90.
1. There are 3 dierent classes: cfw_e, cfw_(1 2), (1 3), (2 3), cfw_(1 2 3), (1 3 2).
We have that ce = 1, c(1 2) = 3, and c(1 2 3) = 2. And, note that
|S3 | = 6 = 1 + 3 + 2.
4. One way that
Math 4107
HW1 solution
1.5 Let be a natural number > 1. A -adic expansion of a number x N is the expression
x = a0 + a1 + + ar r ,
where r N, ai N and 0 ai < .
(i) Compute a 3-adic expansion of 17.
A 3-adic expansion of 17 is
17 = 2 + 2 3 + 1 32 .
(ii) Pr
Math 4107, Midterm 1, Fall 2005
February 14, 2008
1. (10 points)
a. Dene what it means for a set G to be a group.
b. Dene what it means for a mapping from a group G to a group G to
be a homomorphism. Also, dene what it means for to be an isomorphism.
c. D
The group A5 is simple
February 26, 2008
1
Introduction
Recall that a group G is said to be simple if
H
G = H = cfw_e or H = G.
That is, G has only the trivial normal subgroups cfw_e and G itself.
Clearly, Sn , n 3, is not simple, as
An S n ;
but for n 5
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Shanghai Jiao Tong University
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Math 4107 Homework 2
Xinying Chen
January 20, 2015
1. Chapter 1: 17
Suppose X is a natural number such that x 1 (mod 2), x 2 (mod 3) and x 3 (mod 5).
Here n1 = 2, n2 = 3, n3 = 5. Let N = n1 n2 n3 = 30. We need to nd i , j Z such that
1 n1 + 1
N
N
N
= 21 +
Math 4107
Following are all the exam questions from last time I taught 4107. The most dicult ones
were given as bonus problems. The course used a dierent textbook (Hersteins Topics in
Algebra) covering a slightly dierent set of topics. The exams in this c
Math 4107
HW2 solution
Problem 1.19
The last three tables are actually the same if you permute rows and columns simultaneously
and rename a, b, c. For example if you swap a and b in the second table, you get the third
table. There are only two non-isomorp
Name:
Math 4107 Quiz 0 January 5, 2015
Practice Quiz. This does not count toward grade.
1. Let S be a set. Which of the following is a relation on S?
(a) any function from S to R
(b) any subset of S S
(c) both (a) and (b)
(d) neither (a) nor (b)
2. Suppos
Math 4107 Quiz 1 January 14, 2015
1. Which of the following is not an axiom of a group.
(a) associativity
(b) distributivity
(c) existence of inverses
(d) They are all group axioms.
2. Which of the following is an abelian group?
(a) the set of bijections
Math 4107 Quiz 2 January 26, 2015
1. Which of the following is a group homomorphism from (R, +) to (R, +)?
(a) f (x) = 2x
(b) f (x) = 2x
(c) f (x) = x2
(d) f (x) = log2 (x)
2. Which of the following is a group homomorphism from (R>0 , ) to (R>0 , )?
(a) f
Math 4107 Quiz 3 Feb 2, 2015
Cyclic groups and product groups
Recall: (Z/n) = cfw_x Z/n : gcd(x, n) = 1.
21. Let n be a positive integer 2. Which of the following must be true?
(a) (Z/n) , ) is a cyclic group.
(b) (Z/n, +) is a cyclic group.
(c) Both (a)
Final Exam, Math 4107
April 28, 2008
NO CALCULATORS ARE ALLOWED FOR THIS EXAM!
Instructions. Work any 8 of the following 10 problems.
1. Find integers x and y such that
a. 76x + 47y = 1.
b. 68x 1 (mod 109).
2.
a. Determine the number of permutations of th
Group Actions
Ernie Croot
February 8, 2008
1
Introduction
Given a set X we say that a group G acts on X if we can think of the
elements of G as being permutations on X . So, given g G and x X we
can consider y = g (x). We say that g sends the element x to
Math 4107 Study Sheet for the Final Exam
April 26, 2011
1. Know the basics of set theory, mappings, and properties of the integers,
such as divisibility, gcds, the fundamental theorem of arithmetic, prime
numbers, and congruences. Know how to prove that m
Math 4107, exam 1
September 22, 2009
Each question is worth 20 points. 1. Dene the following terms. a. Normal subgroup. b. Homomorphism. c. Innerautomorphism. 2. List all the elements of order 2 from the group S4 . 3. Find integers k and , satisfying 55k
Math 4107, Midterm 2, Fall 2009
November 12, 2009
1. Dene the following terms. a. Ring (list the axioms). b. Centralizer of an element g in a group G. c. State the three Sylow theorems. d. Cayleys Theorem. e. Alternating group. 2. Express the following pe
Selected solutions to Math 4107 midterm 2, Fall 2009
December 11, 2009
1. You can look these up. 2. This is a routine computation, so I wont bother to do it. 3. The center of S2 is all of S2 , since S2 is cyclic of order 2, which is abelian. We claim that
Syllabus for Math 4107, Abstract Algebra
August 18, 2009
Instructor: Ernie Croot email: [email protected]/ecroot Please resist the urge to email me unless it is absolutely necessary. Course Webpage: www.math.gatech.edu/ecroot Click on the Math 4107 l
Math 4107 Midterm 1 Solutions, Fall 2009
October 18, 2009
1. You can look up these denitions yourself.
2. There are many ways to solve this problem. We present two here:
Way 1. Let H be the subgroup of order 35. By Lagrange, we know that
every element of
Study sheet for Math 4107 midterm 2, Fall
2009
November 10, 2009
Know all the material covered up to and including the rst midterm.
Know what an automorphism and inner-automorphism are, and know
that they form a group under composition. Know that the gr
Math 4107, Midterm 1, Fall 2005
September 22, 2009
1. (10 points) a. Dene what it means for a set G to be a group. b. Dene what it means for a mapping from a group G to a group G to be a homomorphism. Also, dene what it means for to be an isomorphism. c.
Some advice to undergraduate math majors
August 22, 2011
1
Introduction
The advice below is sort of a list of suggestions of dos and donts that I
thought I would write, to rectify what I think are unwise decisions that I see
undergraduate math majors make
An is generated by the 3-cycles
March 5, 2008
Here I give a proof of one of your homeworks on An being generated by
the 3-cycles.
The idea of the proof is that we will show that any product of two transpositions is a product of 3-cycles; and, if so, then
Math 4107 Study Sheet for the Final Exam
April 26, 2008
1. Know the basics of set theory, mappings, and properties of the integers, such as divisibility, gcds, the fundamental theorem of arithmetic, prime
numbers, and congruences. Know how to prove that m