MATH 5401 - Abstract Algebra I

1.1

# 5,8,11,12,25,28

- Sam's solutions

- Extra solution to 28

1.2

1(b,c), 4

A) Compute the group table for D_6 and D_8.

B) Find the inverse of r^2 in D_6, r in D_8, r^2 in D_8, and sr in D_8.  What is the inverse of r in D_2n?  What is the inverse of sr^i in D_2n?

C) In D_8, simplify the following products:  s r^2 s r^3 and r s r^(-2).

- Sam's solutions

4550 HW

Do these problems from MATH 4550:

HW 1 - 1, 4, 5, 7, 12, 14

1.3

1, 2, 4(a)

- Sam's solutions

4550 HW

Do these problems from MATH 4550:

HW 3: 1, 2, 4, 5

1, 2, 3, 4, 5, 6, 13, 15

- Sam's solutions

- Extra solutions to 1,3,4,5,6,15 are here.   Extra solutions to 2 and 13 are here.

1.7

14, 15, 16, 17, 18

- Sam's solutions

- Extra solutions to 14, 15, 16 are here.   Extra solutions to 17 and 18 are here.

Do these problems from MATH 4550:

HW 5: 1, 2, 3

2.1

1(b), 2(a), 3(b), 10(a), 11, 12(b), 14

- Sam's solutions

- Extra solution to 11 is here.  Extra solutions to the problems except for 11 are here.

2.2

2, 5(a), 6, 7, 11

- Sam's solutions

- Other solutions are here.

2.3

11, 12, 13, 26

Note: For #26, in the book for this problem, Z_n denotes an abstract cyclic group of order n.  So the notation x^n does not mean multiplication in the group Z_n = {0,1, ..., n - 1}, it means multiplication in the abstract cyclic group.
 

Also, do: 
A) Find all the generators for Z_8 where Z_8 = {0,1,2,3,4,5,6,7} is the integers modulo 8 under addition.

- Sam's solutions

- Other solutions are here.

4550 HW

Do these problems from MATH 4550:

3.1

1, 3, 4, 5, 20, 22(a), 36

A) Calculate the elements of ( Z_2 x Z_4 ) / < (0,1) >
B) Calculate the elements of  D_8 / <s>
C) Give the order of the element (2,1) + <(1,1)> in the factor group (Z_3 x Z_6) / <(1,1)>.

3.2

4, 5, 8, 16

- Sam's solutions

- Extra solutions for the problems (except for 5) are here.   Extra answer to 5 is here.

3.3

For A and B, find groups that the following are isomorphic to and use the first isomorphism theorem to prove it:
A) ( Z_2 x Z_4 ) / < (0,1) >
B) ( Z x Z ) / < (1,2) >

C) Let G and G' be groups, and let H and H' be normal subgroups of G and G', respectively.  Let f be a homomorphism from G to G'.  Show that f induces a natural homomorphism 
                         g : (G / H) --> (G' / H')
if f(H) is a subset of H'

- Sam's solutions

- Extra solutions here

3.5

2

- Sam's solutions

- Extra solutions here

4550 HW

Do these problems from MATH 4550:

TEST 2 covers up to this point

4.3

2(a), 4, 8

- Sam's solutions

- Extra solutions here (Note: there is a typo in the solution for #8 at the very end.  It should say sigma o tau /= tau o sigma.    Right now it has sigma o tau = tau o sigma.)

4.4

2, 15

- Sam's solutions

- Extra solutions here

4.5

5, 13, 30

Hint for 5: Show that <r^{n/p^k}> is the unique Sylow p-group of size p^k if p is odd and 2n = 2p^k * m (where p does not divide m).

- Sam's solutions

- Extra solutions for 5 and 30 are here.   Extra answers to 13 are here

5.2

A) Let G be a finite abelian group.  Prove that G is simple if and only if G is isomophic to Z_p for some prime p.

B) Find all abelian groups of size 36.  Find all abelian groups of size 540. 

- Sam's solutions

- Extra solution to A is here

5.4

A) Classify all groups of size 5^2 * 7 

- Sam's solutions

- Extra solution here