MATH 3E03 Midterm #2 Solutions
Matt Valeriote
Midterm Test
Duration of test: 60 minutes
McMaster University
November 19, 2014
First Name:
Last Name:
Student No.:
Please answer all ve questions. For all questions, write your answers up in
the answer bookle
MATH 3E03 Assignment #6 Solutions
Due: Thursday, November 29, in class
1. Suppose that H and K are normal subgroups of the group G such that
G = HK. Show that G/(H K) is isomorphic to (G/H) (G/K).
Hint: Use the First Isomorphism Theorem after rst nding a
MATH 3E03 Assignment #1 Solutions
Due: Monday, September 22, in class
Important note: All work submitted for grading must be your own. You
may discuss homework problems and related material with other students,
but you must not submit work copied from oth
MATH 3E03 Assignment #4 Solutions
Due: Monday, November 10, in class
1. Let H1 , H2 , K1 , and K2 be groups such that H1 is isomorphic to K1 and
H2 is isomorphic to K2 . Show that H1 H2 is isomorphic to K1 K2 .
Solution: Since Hi is isomorphic to Ki , for
MATH 3E03 Assignment #3 Solutions
Due: Monday, October 20, in class
1. For each pair of group G and subgroup H, describe the left and right
cosets of H in G and nd [G : H]:
(a) G = Q, H = Z.
Solution: Each coset of Z in Q is of the form q + Z for some
rat
MATH 3E03 Midterm #1 SOLUTIONS
Matt Valeriote
Midterm Test
Duration of test: 60 minutes
McMaster University
October 24, 2014
Initials:
Last Name:
Student No.:
Please answer all six questions. To receive full credit, provide justications
for your answers.
A statement in logic or mathematics is an assertion that is either true or false.
A mathematical proof is nothing more than a convincing argument about the accuracy of
a statement
A statement could be as simple as \10=5 = 2"
However, mathematicians are us
MATH 3E03 Assignment #5 Solutions
Due: Monday, November 17, in class
Important note: All work submitted for grading must be your own. You
may discuss homework problems and related material with other students,
but you must not submit work copied from othe
MATH 3E03 Assignment #6 Solutions
Due: Monday, December 1, in class
Important note: All work submitted for grading must be your own. You
may discuss homework problems and related material with other students,
but you must not submit work copied from other
Abstract Algebra
Theory and Applications
Abstract Algebra
Theory and Applications
Thomas W. Judson
Stephen F. Austin State University
Sage Exercises for Abstract Algebra
Robert A. Beezer
University of Puget Sound
August 9, 2016
Website: abstract.pugetsoun
MATH 3E03 Assignment #2 Solutions
Due: Monday, October 6, in class
1. Let G be a group with group operation and let c G be an element
of G. For this choice of c, dene to be the following operation on G:
for x, y G,
x y =xcy
where the product on the right
MATH 3E03 Midterm #2 Solutions
Matt Valeriote
Midterm Test
Duration of test: 60 minutes
McMaster University
November 19, 2013
First Name:
Last Name:
Student No.:
Please answer all ve questions. For all questions, write your answers up in
the answer bookle
MATH 3E03 Assignment #3
Due: Thursday, October 11, in class
1. Give an example of a group of order 60 that is not abelian, and an
element from that group that has order 5. Can a group of order 60
have an element of order 8?
2. Let H and K be subgroups of
Note: In the
original version,
this was 96.
MATH 3E03 Assignment #6
Due: Thursday, November 29, in class
1. Suppose that H and K are normal subgroups of the group G such that
G = HK. Show that G/(H K) is isomorphic to (G/H) (G/K).
Hint: Use the First Isom
MATH 3E03 Assignment #5 Solutions
Due: Thursday, November 1, in class
1. For each group G and subset H, determine if H is a normal subgroup
of G:
(a) G = A4 , H = cfw_(1), (12)(34), (13)(24), (14)(23).
Solution: Since |A4 | = 12 and |H| = 4, then H has th
MATH 3E03 Assignment #5
Due: Thursday, November 1, in class
1. For each group G and subset H, determine if H is a normal subgroup
of G:
(a) G = A4 , H = cfw_(1), (12)(34), (13)(24), (14)(23).
(b) G = D4 , H = cfw_id, s.
(c) G = GL4 (R), H = SL4 (R).
2. De
MATH 3E03 Assignment #4
Due: Thursday, October 25, in class
1. For each pair of groups G and H, determine if they are isomorphic:
(a) G = Q, H = C.
Solution: Since the cardinality of Q is smaller than the cardinality
of C (Q is countably innite while C is
MATH 3E03 Assignment #4
Due: Thursday, October 25, in class
1. For each pair of groups G and H, determine if they are isomorphic:
(a) G = Q, H = C.
(b) G = U (10), H = U (12).
(c) G = Z, H = M3 (Z).
(d) G = Z15 and H = Z3 Z5 .
2. Show that there are exact
MATH 3E03 Assignment #2
Due: Thursday, October 4, in class
NOTE: A Graduate Studies Information Session will be held on Wednesday,
3 October, 4:30pm in HH217.
1. Let G = Z Z. Dene a binary operation
on G as follows:
(a, b) (c, d) = (a + c, (1)c b + d).
(a
MATH 3E03 Assignment #1
Due: Thursday, September 20, in class
Important note: All work submitted for grading must be your own. You
may discuss homework problems and related material with other students,
but you must not submit work copied from others.
1.
MATH 3E03 Assignment #3 Solutions
Due: Thursday, October 11, in class
1. Give an example of a group of order 60 that is not abelian, and an
element from that group that has order 5. Can a group of order 60
have an element of order 8?
Solution: The dihedra
MATH 3E03 Assignment #2
Due: Thursday, October 4, in class
NOTE: A Graduate Studies Information Session will be held on Wednesday,
3 October, 4:30pm in HH217.
1. Let G = Z Z. Dene a binary operation
on G as follows:
(a, b) (c, d) = (a + c, (1)c b + d).
(a
MATH 3E03 Assignment #1 Solutions
Due: Thursday, September 22, in class.
Important note: All work submitted for grading must be your own. You
may discuss homework problems and related material with other students,
but you must not submit work copied from
MATH 3E03 Assignment #2
Due: Thursday, October 6, in class.
1. Let G = Z Z. Define a binary operation on G as follows:
(a, b) (c, d) = (a + c, (1)c b + d).
(a) Show that G with the operation is a group.
Solution:To show that (G, ) is a group we need to sh
Solutions to Math 3E03 Homework 6
2.116. Since D8 and Q are the only nonabelian groups, they are non-isomorphic to
the remaining three. From problem 2.87, D8 and Q are non-isomorphic to each other. It
remains to check that I8 , I4 I2 and I2 I2 I2 are mutu
Solutions to Math 3E03 Homework 5
2.96. We shall show the rst isomorphism U (I9 ) I6 . Note that the group U (I9 ) =
=
cfw_[1], [2], [4], [5], [7], [8] has order 6. Also one sees that [2]2 = [4], [2]3 = [8] and [2]6 = [1].
Thus [2] has order 6 and since U
MATH 3E03 Midterm #1 Solutions
Midterm Test
Duration of test: 50 minutes
McMaster University
October 26, 2016
Matt Valeriote
First Name:
Last Name:
Student No.:
Please answer all five questions. To receive full credit, provide justifications
for your answ
MATH 3E03 Assignment #3
Due: Monday, October 24, in class.
Important note: All work submitted for grading must be your own. You
may discuss homework problems and related material with other students,
but you must not submit work copied from others.
1. Let
MATH 3E03 Assignment #4 Solutions
Due: Wednesday, November 9, in class.
1. (a) Is Z60 isomorphic to Z2 Z2 Z15 ?
Solution: The highest possible order of an element in Z2 Z2 Z15 is lcm(2, 2, 15) =
30. Since Z60 has an element of order 60, these groups are n
MATH 3E03 Assignment #2 Solutions
1. Let G denote the set of non-zero real numbers and consider the following binary operation on G:
ab if a > 0
ab=
.
a/b if a < 0
Determine if G along with the operation is a group. Justify your
answer.
Solution: We need
MATH 3E03 Assignment #4
Due: Thursday, October 27, in class
1. For each pair of groups G and H, determine if they are isomorphic:
(a) G = Z, H = 4Z.
Solution: These groups are both innite cyclic groups, Z is generated by 1, and 4Z is generated by 4, and s