HW 1, MAT 312/AMS 351, LECTURE 01: SPRING 2016
Do problems #3,6 in section 1.1, as well as the following problems.
Problem I: Find the greatest common divisor (gcd) of the numbers 210
and 56; write this gcd in the form 210m + 56n for integers m,n.
Problem
MAT 312/AMS 351
Final Exam.
December 12, 2012
5:30pm8:00pm
Name:
ID:
DO NOT OPEN THIS EXAM YET
(1) Fill in your name and Stony Brook ID number.
(2) This exam is closed-book and closed-notes; no calculators, no phones.
(3) Please write legibly to receive c
HW 7, MAT 312/AMS 351: SPRING 2016
Show some work or explain your reasoning, for your solutions of each of
the following prolems.
In problems (1)-(3) below define permutations on 9 letters , S(9) to
be the following products of cycles: = (13869)(27831)(43
MAT 312/AMS 351, LECTURE 02: SPRING 2016
Topics and Text:
In this course we will discuss elementary results in number theory, set theory and group theory. Also several topics in coding theory are developed,
related to number theory and group theory Our te
ANOTHER REVIEW FOR FINAL EXAM; MAT 312/ AMS
351 FALL 2013
(1) Dene a permutation S6 by = (1, 2, 3)(2, 1)(3, 6)(3, 4, 6). Write
as a product of disjoint cycles. Is the cyclic group < > isomorphic to Z6 ?
(2) Let D5 denote the symmetry group of the pentago
HW 1, MAT 312/AMS 351, LECTURE 02: SPRING 2016
Do problems #3 and #6 in section 1.1, as well as the following problems:
Problem I: Find the greatest common divisor (gcd) of the numbers 210
and 56; write this gcd in the form 210m + 56n for integers m, n.
P
SOLUTIONS FOR REVIEW FOR MIDTERM FOR MAT
312/AMS 351: SPRING 2016
The midterm will cover sections 1.1-1.6, 2.1-2.3, 4.1.
(1) Let a,b,p,q denote integers which satisfy: a = bp + 756; b = 756q + 315.
Using only this information compute the greatest common d
HOMEWORK WEEK 4
05 AUGUST 2014
Deadline: 12 August 2014, TUESDAY 13:00. (Notice the change of day.)
Q. 1. Show whether the groups G10 and G7 are cyclic or not. If so, determine their generators.
Proof. The subgroups generated by the elements of G7 are
[1]
MAT 312 - AMS 351
FINAL EXAM
SUMMER II, 14 August 2014
Question 1.
a. Find all four solutions to the equation x2 1 0 mod 35.
(5 pts)
b. Solve the equation [243]n [x]n [1]n for n = 1130.
(7 pts)
c. Find the last three digits of the integer
2001
20032002
.
MAT 312/AMS 351
Solutions to Final Exam.
1. (30 points) a) (10 points) Solve the system of congruences
x = 10
mod 13,
x = 16
mod 19.
mod 13, x = 3
mod 19,
Answer:x = 244 mod 247.
Solutions: 1) Remark that
x = 3
so
x = 3
mod 13 19,
x = 244
mod 247.
2) Sinc
AMS 311, Fall 2014
Homework 3
Due at the beginning of class, Sept 23, 2014
1. (20 points, 5 points each) If P (A) = 0.8, P (B|A) = 0.5, A and B are independent, for
each statement, say True, False, or Cant Tell (if you give some reasoning, you have
a poss
AMS 311, Fall 2014
Homework 2: Suggested Solution Key
1. Sarah has n keys, of which one will open the door.
(a) If she tries the keys at random, discarding those that do not work, what is the
probability that she will open the door on the kth try?
p = P (
AMS 311, Fall 2014
Homework 4
Due at the beginning of class, Oct 7, 2014
1. (20 points) A company is deciding on installing a 3-component or 5-component satellite
system in XYZ town. It is known that the system functions on any given day if a majority
of
AMS 311, Fall 2014
Homework 2
Due at the beginning of class, Sept 16, 2014
1. (10 points, 5 points each) Sarah has n keys, of which one will open the door.
(a) If she tries the keys at random, discarding those that do not work, what is the
probability tha
AMS 311, Fall 2014
Homework 3: Suggested Solution Key
1. (20 points, 5 points each) If P (A) = 0.8, P (B|A) = 0.5, A and B are independent, for
each statement, say True, False, or Cant Tell (if you give some reasoning, you have
a possibility of partial cr
MAT 312 / AMS 351 Applied Algebra
Stony Brook University
Exam 1, Thursday 10/27/2016
Jason Starr
Fall 2016
MAT 312 / AMS 351 Exam 2 Review
Exam Policy. Exam 1 will be held on Thursday, October 27th, during lecture. The exam is closed
book, closed notes, n
Homework #11 Solutions
Problem 1
Problem 1
Let G be a group. Show that for each element a G, ( a1 )1 = a.
Solution
Note that a ( a1 ) = e and that ( a1 ) ( a1 )1 = e. We have
a = a e = a ( a 1 ) ( a 1 ) 1 = ( a 1 ) 1
Problem 2
Let ( G, ) be a group. Show
MAT 312/AMS 351 Homework 12 Solutions
Giulia Sacca, Yuanqi Wang, Benjamin Sokolowsky
May 6, 2017
Throughout this assignment we will use the subgroup criterion (iii) for a nonempty set H, that
x, y H implies xy 1 H. One could alternatively show that H is c
HOMEWORK 10 SOLUTIONS, MAT 312/AMS 351, SPRING
2017
Problem 1: Solution Part a. (15)1031 = (15)2515+1 = (15)2 )515 (15) =
id515 (15) = (15).
Part b. (285)277 = (285)392+1 = (285)3 )92 (285) = id92 (285) = (285).
Problem 2: Solution Part a. = (1623)(3245)
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MAT 312 / AMS 351 Applied Algebra
Stony Brook University
Problem Set 5, Due Thursday 10/13/2016
Jason Starr
Fall 2016
MAT 312/AMS 351 Problem Set 5
Homework Policy. Write up solutions of the required problems. Read and attempt the extra
problems, but do n
MAT 312 / AMS 351 Applied Algebra
Stony Brook University
Problem Set 10, Due Thursday 12/01/2016
Jason Starr
Fall 2016
MAT 312/AMS 351 Problem Set 10
Homework Policy. Write up solutions of the required problems. Read and attempt the extra
problems, but do
MAT 312 / AMS 351 Applied Algebra
Stony Brook University
Problem Set 7, Due Thursday 11/3/2016
Jason Starr
Fall 2016
MAT 312/AMS 351 Problem Set 7
Homework Policy. Write up solutions of the required problems. Read and attempt the extra
problems, but do no
MAT 312 / AMS 351 Applied Algebra
Stony Brook University
Problem Set 1, Due Thursday 9/8/2016
Jason Starr
Fall 2016
MAT 312/AMS 351 Problem Set 1
Homework Policy. Write up solutions of the required problems. Read and attempt the extra
problems, but do not
MAT 312 / AMS 351 Applied Algebra
Stony Brook University
Problem Set 10, Due Thursday 12/08/2016
Jason Starr
Fall 2016
MAT 312/AMS 351 Problem Set 11
Homework Policy. Write up solutions of the required problems. Read and attempt the extra
problems, but do
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MAT 312 / AMS 351 Applied Algebra
Stony Brook University
Problem Set 2, Due Thursday 9/15/2016
Jason Starr
Fall 2016
MAT 312/AMS 351 Problem Set 2
Homework Policy. Write up solutions of the required problems. Read and attempt the extra
problems, but do no
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Contents
Preface to rst edition
Preface to second edition
Introduction
Advice to the reader
1
1.1
1.2
1.3
1.4
1.5
1.6
2
2.1
2.2
2.3
2.4
3
3.1
3.2
3.3
4
4.1
4.2
4.3
page ix
x
xi
xiv
Number theory
The division algorithm and greatest common divisors
Mathemat
4 Examples of groups
The mathematical concept of a group unies many apparently disparate ideas.
It is an abstraction of essential mathematical content from particular situations.
Abstract group theory is the study of this essential content. There are seve
5 Group theory and
error-correcting codes
By now we have met many examples of groups. In this chapter, we begin by
considering the elementary abstract theory of groups. In the rst section we
develop the most immediate consequences of the denition of a gro
2 Sets, functions and
relations
In this chapter we set out some of the foundations of the mathematics described
in the rest of the book. We begin by examining sets and the basic operations on
them. This material will, at least in part, be familiar to many