Solutions to Homework 7
3.1, 1 Using ordinary addition of integers as the operation, show that the
set of even integers forms a group but that the set of odd integers
does not.
Proof: Addition is an associative binary operation on Z. The
sum 3 + 5 = 8 is
HW6 solutions (3.2, 3.3)
3.2.5
Math 145, Abstract Algebra, Duchin
Find all cyclic subgroups of. (b) Z8 ; (d) S4
For Z8 : We have 0 = cfw_0. For anything relatively prime to 8, it generates the
full group, so we have 1 = 3 = 5 = 7 = Z8 . We know that a gen
Math 145-02
Tufts University Dept of Mathematics
Final Exam
Dec 15, 2011
Prof. McNinch
Throughout this exam, the symbol Z represents the set of all integers, Q represents the set of all
rational numbers, and C represents the set of all complex numbers. If
Math 145-02
Tufts University Dept of Mathematics
Final Exam Review Material
Dec 12, 2011
Prof. McNinch
The nal in Math 145 is comprehensive, so you are responsible for all of the material covered in the
course. However, the nal will emphasize the material
Math 145-02
Tufts University Dept of Mathematics
Final Exam Review Material
Dec 13, 2013
Prof. McNinch
The nal in Math 145 is comprehensive, so you are responsible for all of the material covered in
the course. However, the nal will emphasize the material
Math 145-02
Tufts University Dept of Mathematics
Exam 1 Review Material
Oct 3, 2011
Prof. McNinch
Throughout this exam, the symbol Z represents the set of all integers, and Q represents the set of all
rational numbers. If n Z>0 , then Zn denotes the set o
Solutions to Homework 1
1.1, 3(c) Computational. I will do not do this out here.
1.1, (7) Let a, b, c Z. Give proofs of the following:
(a) If b|a, then b|ac.
Proof: Since b|a, we know a = bq for some integer q. Then using
associativity of multiplication w
Solutions to Homework 5
1.2, (8) Let a, b be positive integers with d = (a, b). Since d|a and d|b,
there exists integers h, k such that a = dh and b = dk. Show that
(h, k) = 1.
Proof: We know d can be written as am + bn for some m, n Z. We
can rewrite thi
Solutions to Homework 6
1.3, (13) Prove that the sum of the cubes of any three consecutive positive
integers is divisible by 9.
Proof: We must show that for any integer a, we have: (a 1)3 +
a3 + (a + 1)3 0 (mod 9)
To do this, we compute:
(a1)3 +a3 +(a+1)3
Math 145, Abstract Algebra, Duchin
Quiz 1
The grading of this quiz will focus on clear argumentation.
(a) Clearly prove that b a,
b (a + c) = c bZ.
Proof. The denition of bZ is the set of all integer multiples of
b, so x bZ b x x = bk for some k Z. So I
w
Quiz 2
Math 145, Abstract Algebra, Duchin
(1) Give an example of a well-dened function from Z10 Z3 that is neither
injective nor surjective. (Or if none exists, say why not.)
The constant map dened by f (x) = [2]3 for all x Z10 is certainly
well-dened bec
Math 145-02
Tufts University Dept of Mathematics
Exam 2 Review Material
Oct 3, 2011
Prof. McNinch
Throughout this exam, the symbol Z represents the set of all integers, and Q represents the set of all
rational numbers. If n Z>0 , then Zn denotes the addit
Math 145-02
Tufts University Dept of Mathematics
Exam 1
Oct 3, 2011
Prof. McNinch
Throughout this exam, the symbol Z represents the set of all integers, and Q represents the set of all
rational numbers. If n Z>0 , then Zn denotes the set of congruence cla
Sec 3.1 problems from HW3
3.1.2
Math 145, Abstract Algebra, Duchin
For each binary operation , does the set with dene a group?
(a),(c),(e) are in the back of the book, so Ill just do the others. Note: I will often
use the letter e for the identity element
Quiz 3
Math 145, Abstract Algebra, Duchin
(1) Prove that the order of any group element is equal to the order of its inverse.
By denition of order, o(a) = | a |, and by denition of generation,
a = cfw_an : n Z.
Note that cfw_an : n Z = cfw_an : n Z, becau