Math 4124
Monday, February 14
February 14, Ungraded Homework
Exercise 3.1.22(a) on page 88 Prove that if H and K are normal subgroups of a group G,
then their intersection H K is also a normal subgroup of G.
We have already proved previously that H K is a
Math 4124
Wednesday, January 26
First Homework Solutions
1. 1.1.9(b) on page 22 Let G = cfw_a + b 2 R | a, b Q. Prove that the nonzero
elements of G form a group under multiplication.
If x, y G then we may write x = + b 2 and y = c + d 2 where a, b, c, d
Math 4124
Wednesday, January 19
January 19, Ungraded Homework
Exercise 1.1.5 on page 21 Prove for all n > 1 that Z/nZ is not a group under multiplication of residue classes.
Z/nZ = cfw_0, . . . , n 1. Suppose Z/nZ is a group. Then it must have an identity
Math 4124
Monday, March 21
Seventh Homework
Due 2:30 p.m., Monday March 28
1. Section 4.4, Exercise 3 on page 137. (If Aut(D8 ), show (r2 ) = r2 , and deduce
(s) = r2 .)
(3 points)
2. Let G be a group. If Aut(G) is cyclic, prove that G is abelian. Hint:
Math 4124
Monday, March 14
Sixth Homework
Due 2:30 p.m., Monday March 21
1. Section 4.2, Exercise 2 on page 121.
(3 points)
2. Let N S5 , and suppose N = 1 or S5 .
(a) Determine the conjugacy classes of S5 and their size.
(b) Show that N cannot contain a
Math 4124
Wednesday, February 23
Fifth Homework
Due 2:30 p.m., Monday March 14
1. Let G be a group of order 10 with a normal subgroup H of order 2.
(a) Prove that H is contained in the center of G (hint: If 1 = x H and g G, then
1 = gxg1 H ).
(b) Prove th
Math 4124
Monday, February 7
Fourth Homework
Due 2:30 p.m., Monday February 14
1. Section 2.3, Exercise 6 on page 60.
(3 points)
2. Section 2.4, Exercise 3 on page 65. (Use an ungraded homework problem.) (2 points)
3. Let G be the group of all rigid motio
Math 4124
Monday, January 31
Third Homework
Due 2:30 p.m., Monday February 7
1. Section 1.5, Exercise 2 on page 36, just do Q8 .
(3 points)
2. Section 1.6, Exercise 7 on page 40. (From the previous problem, Q8 has exactly one
element of order 2.)
(2 point
Math 4124
Monday, January 24
Second Homework
Due 2:30 p.m., Monday January 31
1. Section 2.1, Exercise 10(a) on page 48
(2 points)
2. Section 1.2, Exercise 3 on page 27.
(2 points)
3. Section 1.2, Exercise 10 on page 28. (A face can be moved to any of the
Math 4124
Monday, February 28
February 28, Ungraded Homework
Exercise 3.5.11 on page 111 Prove that S4 has no subgroup isomorphic to Q8 .
In Q8 we have the relation i j = k, and of course i, j, k all have order 4; in other words we have
the product of two
Math 4124
Wednesday, February 23
February 23, Ungraded Homework
Exercise 3.3.4 on page 101 Let C be a normal subgroup of the group A and let D be a
normal subgroup of the group B. Prove that (C D) (A B) and (A B)/(C D)
=
(A/C) (B/C).
Dene : A B (A/C) (B/
Math 4124
Wednesday, February 9
February 9, Ungraded Homework
Exercise 3.1.4 on page 85 Prove that in the quotient group G/N , (gN ) = g N for all
Z.
If is positive, this is proved by induction. The result is certainly true if = 0 (both sides
are then th
Math 4124
Monday, February 7
February 7, Ungraded Homework
Exercise 2.3.12 on page 60 Prove that the following groups are not cyclic.
(a) Z/2Z Z/2Z
(b) Z/2Z Z
(c) Z Z
(a) |Z/2Z Z/2Z| = 4, so if the group was cyclic, it would have an element of order 4.
Th
Math 4124
Monday, April 18
Ninth Homework
Due 2:30 p.m., Monday April 25
1. Section 7.3, Exercise 10 (a),(b),(c),(d) on page 248.
(3 points)
2. Section 7.3, Exercise 34 on page 250. (For (d) write 1 = i + j and if x I J , consider
x = xi + x j.)
(3 points
Math 4124
Monday, April 25
Tenth Homework
Due 2:30 p.m., Monday May 2
1. Let R be a commutative ring with a 1 = 0, and let S denote the set of nonzerodivisors
of R (that is cfw_s R | sr = 0 for all r R \ 0).
(a) Prove that S is a multiplicatively closed s
Math 4124
Wednesday, February 16
First Test Review
The test will cover up to (and including) section 3.2. Topics will include
1. The order of an element. If n and r are positive integers and |g| = n, then |gr | =
n/(n, r).
2. Direct product of two groups
Math 4124
Wednesday, January 19
Policy Sheet
Course
Prereq
Instructor
Book
Ofce
Telephone
E-mail
Room
Math 4124, Introduction to Abstract Algebra CRN (index number) 14253
Math 3124 (Modern Algebra), VERY IMPORTANT!
Peter A. Linnell
Abstract Algebra 3rd. e
Math 4124
Monday, May 2
May 2, Ungraded Homework
Prove that 2 is irreducible but not prime in Z[ 13].
Here Z[ 13] = cfw_a + b 13 | a, b Z. Dene N : Z[ ] Z by N (a + b 13) = a2 13b2
13
to
for a, b Z. Then N ( ) = N ( )N ( ) for , Z[ 13]; see this write = a
Math 4124
Monday, March 28
March 28, Ungraded Homework
Prove that a group of order 765 is abelian.
Let G be a group of order 765 = 9 5 17. We need to prove that G is abelian. It is
usually best to consider the largest prime dividing the order of the group
Math 4124
Wednesday, March 23
March 23, Ungraded Homework
Exercise 4.5.18 on page 147. Prove that a group of order 200 has a normal Sylow 5subgroup.
200 = 8*25. Therefore the number of Sylow 5-subgroups divides 8 and is congruent to 1
mod 5. The only poss
Math 4124
Monday, March 21
March 21, Ungraded Homework
Prove that a group of order 175 is abelian.
Let G be the group of order 175 = 7 52 . The number of Sylow 7-subgroups is congruent
to 1 mod 7 and divides 25. The only possibility is 1, so G has a norma
Math 4124
Wednesday, March 16
March 16, Ungraded Homework
Exercise 4.4.1 on page 137 Prove that if Aut(G) and g is conjugation by g, then
g 1 = (g) . Deduce that Inn(G) Aut(G).
Let x G. Then
g 1 (x) = (g( 1 (x)g1 ) = ( g)( 1 (x)( g)1 = (g) (x).
Since th
Math 4124
Monday, March 14
March 14, Ungraded Homework
Exercise 4.2.1 on page 121 Let G = cfw_1, a, b, c be the Klein 4-group.
(a) Label 1, a, b, c with the integers 1,2,4,3, respectively, and prove that under the left regular
representation of G into S4
Math 4124
Wednesday, March 2
March 2, Ungraded Homework
Exercise 4.1.4 on page 116 Let S3 act on the set of ordered pairs: cfw_(i, j) | 1 i, j 3
by (i, j) = ( (i), ( j). Find the orbits of S3 on . For each S3 nd the cycle
decomposition of under this actio
Math 4124
Monday, January 31
January 31, Ungraded Homework
Exercise 2.2.1 on page 52 Prove that CG (A) = cfw_g G | g1 ag = a for all a A.
By denition, g CG (A) if and only if gag1 = a for all a A. Multiplying on the left
by g1 and on the right by g, we se
Math 4124
Wednesday, January 26
January 26, Ungraded Homework
Exercise 1.3.1 on page 32 Let be the permutation
13
24
35
42
51
23
32
44
5 1.
and let be the permutation
15
Find the cycle decomposition of each of the following permutations , , 2 , , , and
2
Math 4124
Monday, January 24
January 24, Ungraded Homework
Exercise 1.2.1 on page 27 Compute the orders of each of the elements in the following
groups.
(a) D6
(b) D8
(c) D10
We shall only give details for D10 .
(a) 1 has order 1; r, r2 have order 3; and
Math 4124
Wednesday, February 2
February 2, Ungraded Homework
Exercise 2.3.1 on page 60 Find all subgroups of Z45 = x , giving a generator for each.
Describe the containments between these subgroups.
The problem is equivalent to nding all the subgroups an