Modern Algebra 1
Spring 2010
Homework 1.1
Due January 20
Exercise 1. Let G be a nonempty set with a binary operation. We say G is a monoid if:
(i) a(bc) = (ab)c for all a, b, c G and (ii) there is an e G (called an identity) so that
ae = ea = a for all a
Modern Algebra 1
Spring 2010
Homework 10.1
Due April 7
Exercise 1. Let G be a group and let H, K G. Prove that if K H and G/K is cyclic,
then G/H is cyclic. [Suggestion: Use the Third Isomorphism Theorem and the fact (proven
in earlier homework) that a qu
Modern Algebra 1
Spring 2010
Homework 11.2
Due April 14
Exercise 4. Determine if the permutations in Exercise 1 are even or odd.
Exercise 5. Prove that a cycle in Sn is even if and only if its length is odd.
Exercise 6. Let f (x1 , x2 , . . . , xn ) be a
Modern Algebra 1
Spring 2010
Homework 12.1
Due April 21
Exercise 1. How many abelian groups are there of order n if n = 2008, 2009, 2010 or 2011?
Exercise 2. Let G be an abelian group (written additively). Recall that for m N we
dened
Gm = cfw_x G | mx =
Modern Algebra 1
Spring 2010
Homework 12.2
Due April 21
Exercise 3. Let f : G H be a homomorphism of abelian groups. We say that f splits
if there is a homomorphism g : H G so that f g = IdH . Show that if f splits then
G ker f Im f . [Hint: Dene F : ker
Modern Algebra 1
Spring 2010
Homework 12.3
Due April 21
Exercise 5. Let G be an (additive) abelian group and let m Z.
a. Prove that the function fm : G G given by fm (x) = mx is a homomorphism.
b. Use part a to show that Gm and mG = cfw_mx | x G are both
Modern Algebra 1
Spring 2010
Homework 13.1
Due April 30
Exercise 1. Find (with proof!) the period and repeating part of the decimal expansion of
1/239.
Exercise 2. For what natural numbers a does the decimal expansion of 1/a have period
exactly 4? [Hint:
Homework #3 Solutions
p 23, #4. s = 3 and t = 2 work since
7s + 11t = 21 + 22 = 1.
These choices are not unique since s = 8, t = 5 also work:
7s + 11t = 56 55 = 1.
p 24, #30. Given any integer n, at least one of the three consecutive numbers n 1, n, n + 1
Homework #4 Solutions
p 67, #8. In U (14) we have
32
33
34
35
36
mod 14
mod 14
mod 14
mod 14
mod 15
=
=
=
=
=
9
27 mod 14 = 13
3 13 mod 14 = 39 mod 14 = 11
3 11 mod 14 = 33 mod 14 = 5
3 5 mod 14 = 15 mod 14 = 1
52
53
54
55
56
mod 14
mod 14
mod 14
mod 14
m
Homework #5 Solutions
p 83, #16. In order to nd a chain
a1 a2 an
of subgroups of Z240 with n as large as possible, we start at the top with an = 1 so that
an = Z240 . In general, given ai we will choose ai1 to be the largest proper subgroup
of ai . We wil
Homework #8 Solutions
p 132, #10 () Suppose that is an automorphism of G. Let a, b G. Then
b1 a1 = (ab)1 = (ab) = (a)(b) = a1 b1
which implies that ab = ba. Since a, b G were arbitrary we conclude that G is abelian.
() Suppose that G is abelian. Let a, b
Homework #9 Solutions
p 149, #18 Let n > 1. Then n 1 U (n) and (n 1)2 = n2 2n +1 so that (n 1)2 mod n =
1. Since n 1 = 1, this means that |n 1| = 2 in U (n). As the order of any element in a
group must divide the order of that group, it follows that 2 mus
Homework #11 Solutions
p 166, #18 We start by counting the elements in Dm and Dn , respectively, of order 2. If
x Dm and |x| = 2 then either x is a ip or x is a rotation of order 2. The subgroup of
rotations in Dm is cyclic of order m, and since m is even
Modern Algebra
Practice Exam - Solutions
Disclaimer: This practice exam is not intended to reect the content of Wednesdays
midterm. It is simply a list of problems left over from the preparation of the actual exam,
and should serve to indicate the general
Practice Problem Solutions
1. Despite its appearance, this is a problem dealing exclusively with cosets and not using
Lagranges Theorem. We start with the following observation. Let a, b K and suppose
that a(H K ) = b(H K ). Then a1 b H K H and so aH = bH
Modern Algebra 1
Spring 2010
Homework 9.1
Due March 31
Exercise 1. Let G be a group. Prove that [G : Z (G)] is never prime.
Exercise 2. Let G be a group, H a subgroup, and N a normal subgroup. Prove that if
G/N is abelian then H/(H N ) is abelian.
Modern Algebra 1
Spring 2010
Homework 8.3
Due March 24
Exercise 10. Let f : R R R be given by f (x, y ) = 2x 3y .
a. Prove that f is a surjective homomorphism.
b. Find ker f and describe it and its cosets geometrically.
c. The First Isomorphism Theorem im
Modern Algebra 1
Spring 2010
Homework 8.2
Due March 24
Exercise 5. Let G be a group. Show that if H and K are both normal subgroups of G
and H K = cfw_e then xy = yx for all x H and y K . [Hint: Consider the element
xyx1 y 1 .]
Exercise 6. Let G and H be
Modern Algebra 1
Spring 2010
Homework 1.2
Due January 20
Exercise 4. Let G be a group. Use induction to prove that if a1 , a2 , . . . , an G then
(a1 a2 an )1 = a1 a1 a1 .
2
1
n
Exercise 5. Prove that |Sn | = n! and that Sn is non-abelian if n 3.
Exercise
Modern Algebra 1
Spring 2010
Homework 2.2
Due January 27
Exercise 5. Let G be a group and x, y G. Prove that if xy = yx1 then xn y = yxn for
all n Z. [Note: Induction can be used to prove this for all n N. Dont forget to deal
with the negative integers as
Modern Algebra 1
Spring 2010
Homework 3.1
Due February 3
Exercise 1. What is the largest possible order of an element of S7 ?
Exercise 2. Let Sn and let i cfw_1, 2, . . . , n. We say that i is a xed point of if
(i) = i. Prove that if is a cycle containin
Modern Algebra 1
Spring 2010
Exercise 6. Given a 2 2 matrix A =
Homework 3.2
Due February 3
ab
cd
, recall that its determinant is det A =
ac
. Prove the following properties
bd
of the determinant and the transpose. You can assume that the matrices involv
Modern Algebra 1
Spring 2010
Homework 3.3
Due February 3
Exercise 10. The set GL2 (C) of invertible 2 2 matrices with complex entries can be
shown to be a group under matrix multiplication. In fact, the proof given for matrices
with real entries can be us
Modern Algebra 1
Spring 2010
Homework 4.1
Due February 10
01
0i
and B =
(this
1 0
i0
group is known as the quaternion group, by the way). Find all the subgroups of Q and draw
the subgroup lattice for Q. [Note: It may be useful to recall that Q = cfw_I, A,
Modern Algebra 1
Spring 2010
Homework 4.2
Due February 10
Exercise 5. Let f : G H be a group homomorphism.
a. Let a G. Prove that f (an ) = f (a)n for all n Z.
b. Let a G. Prove that f ( a ) = f (a) .
c. Let a G. Prove that if a has nite order then so doe
Modern Algebra 1
Spring 2010
Homework 5.1
Due February 17
Exercise 1. Let f : G H be an isomorphism of groups.
a. Let a G. Prove that |a| = |f (a)|.
b. Prove that G is abelian if and only if H is abelian.
Exercise 2. Let f : G H be an isomorphism of group
Modern Algebra 1
Spring 2010
Homework 5.2
Due February 17
Exercise 5. Let G be a group and for a, b G dene a b if and only if there is an x G
so that xax1 = b. Prove that is an equivalence relation on G.
Exercise 6. Let n 2 and Sn . Prove that (a1 a2 ak )