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 subgrou
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 w
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 o
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 grou
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
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 (hi
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 pr
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
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
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
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
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.
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 detail
EL MUNDO DE LA EFICIENCIA ENERGTICA
(Una bsqueda urgente por conseguirla)
El Mundo de la Energa
Todos lo das, prcticamente todos
los medios de comunicacin nos
recuerdan el cambio climtico.
Por estos d
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,
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), V
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. T
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 con
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-subgrou
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
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)
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 t
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
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. Mul