Abstract Algebra
Homework Assignment
Week 1
Due 09/14/07

 

Problem 1:  Consider the set of symmetries of  a square.  First,  take a square and label each of the the corners by a letter.   Each symmetry will bring exactly one
of  the labeled corners into the lower left hand position.  Make a table with the following columns :  The letter of the corner describing the symmetry, a description of the action producing this symmetry,  a Greek letter  representing the symmetry,  and a matrix describing the isometry of the plane producing the symmetry.  Then make a complete table of the multiplication of  the symmetries, and a table of the multiplication of the matrices and verify that these two tables correspond.

Problem 2:  Label the four vertices of the square with the numbers 1 through 4.  Describe the permutation of the vertices induced by each of the symmetries of the square.   Make a table showing the products of each of  these permutations,  using cycle notation.  Verify that the products of the permutations correspond to the products of the symmetries.
 

Problem 3:  Suppose that T(x)= Ax+b is an invertible affine transformation of  R^3.  Show that T^{-1} is also an affine transformation,  by computing its formula.
 

Problem 4:  Do problem 1.5.3  on page 24 of the text.
 

Problem 5:  Do problem 1.5.9 on page 25 of the text..
 











 

Abstract Algebra
Homework Assignment
Week 2
Due 09/21/07

Problem 1:  Do problem 1.6.2 on page 35 of the text.

Problem 2:  Do problem 1.6.4 on page 35 of the text.

Problem 3:  Do problem 1.6.6 on page 35 of the text.

Problem 4:  Do problem 1.6.10 on page 35 of the text.

Problem 5:  Do problem 1.6.11 on page 35 of the text.

Problem 6:  Suppose that not both m and n are zero. Let  d= gcd(m, n),  m' = m/d  and  n' = n/d.  Show that  gcd ( m', n' ) = 1.

Problem 7:  Do problem 1.7.1 on page 43 of the text.

Problem 8:  Do problem 1.7.2 on page 43 of the text.

Abstract Algebra
Homework Assignment
Week 3
Due 09/28/07

Problem 1:  Do problem 1.7.4 on page 43 of the text.

Problem 2:  Do problem 1.7.5  on page 43 of the text.

Problem 3:  Do problem 1.7.6 on page 43 of the text.

Problem 4:  Do problem 1.7.8 on page 43 of the text, but only do this for n= 6, 7 and 8.

Problem 5:  Do problem 1.7.14 on page 43 of the text.

Problem 6:  Suppose that  f is an irreducible polynomial.  Prove that it is prime.

Problem 7:  Do problem 1.8.7 on page 54 of the text.

Problem 8:  Do problem 1.8.10 on page 55 of the text.

Abstract Algebra
Homework Assignment
Week 4
Due 10/05/07

Problem 1:  Do problem 1.8.16 on page 56 of the text.

Problem 2:  Show that every nonzero element of  Z_n is either invertible or a zero divisor.

Problem 3:  Do problem 1.9.3 on page 67 of the text.

Problem 4:  For the symmetries of a square, if   r is a rotation countercluckwise by the angle pi/2 and  s is any reflection,   the list of  all
                    symmetries is just   { e ,  r,  r^2,  r^3,  s,  r s,  r^2 s,  r^3 s} . Show that the products of all  symmetries are completely determined by
                     the  formula    s r = r^3 s.

Problem 5:  Do problem 1.9.10 on page 68 of the text.

Abstract Algebra
Homework Assignment
Week 5
Due 10/12/07

Problem 1:  Do problem 1.10.1 on page 73 of the text.

Problem 2:  Do problem 1.10.2 on page 73 of the text.

Problem 3:  Do problem 1.10.3 on page 73 of the text.

Problem 4:   Do problem 1.10.5 on page 74 of the text.

Problem 5:  Do problem 1.10.10 on page 74 of the text.

Problem 6:  Do problem 1.11.1 on page 79 of the text.

Problem 7:  Do problem 1.11.3 on page 79 of the text.

Problem 8:  Do problem 1.11.11 on page 79 of the text.

Abstract Algebra
Homework Assignment
Week 6
Due 10/19/07

Problem 1:  Do problem 2.2.2 on page 100 of the text.

Problem 2:  Do problem 2.2.4 on page 100 of the text.

Problem 3:  Do problem 2.2.6 on page 101 of the text.

Problem 4:   Do problem 2.2.8 on page 101 of the text.

Problem 5:  Do problem 2.2.19 on page 102 of the text.

Problem 6:  Do problem 2.2.22 on page 102 of the text.

Problem 7:  Do problem 2.2.23 on page 102 of the text.

Problem 8:  Do problem 2.2.25 on page 103 of the text.

Abstract Algebra
Homework Assignment
Week 7
Due 10/26/07

Problem  1:  Do problem 2.3.4 on page 108 of the text.

Problem  2:  Do problem 2.3.7 on page 108 of the text.

Problem  3:  Show that the  rotations form a normal subgroup of the symmetry group of the disk.

Problem  4:   Let G be a group Define the automorphism group Aut(G) of  G to be the set of all  isomorphisms  f:  G --> G.
                      Show that  Aut (G) is a subgroup of the group of all permutations of  G.

Problem  5:  Let  G be a group and g be an element of  G.  Define the map  i_g :G --> G by  i_g (x)  =g x g^{-1}. 
                    Show that  i_g is an automorphism of  G .  

Problem  6:  Do problem 2.4.5 on page 116 of the text.

Problem  7:  Do problem 2.4.6 on page 116 of the text.

Problem  8:  Do problem 2.4.14 on page 117 of the text.

Problem  9:  Do problem 2.4.21 on page 116 of the text.

Problem 10:  Suppose that  f : G -- >  G'  is a homomorphism of groups and    H' is a normal subgroup of   G'. 
                      Show that  f^{-1}(H) is a normal subgroup of   G.

Abstract Algebra
Homework Assignment
Week 10
Due 11/09/07

Problem  1:  Do problem 2.5.4 on page 123 of the text.

Problem  2:  Do problem 2.5.13 on page 124 of the text.

Problem  3:  Do problem 2.6.2 on page 132 of the text.

Problem  4:   Do problem 2.6.5 on page 132 of the text.

Problem  5:  Do problem 2.7.1 on page 143 of the text.

Problem  6:  Do problem 2.7.4 on page 143 of the text.

Problem  7:  Do problem 2.7.7 on page 143 of the text.

Problem  8:  Do problem 2.7.8 on page 143 of the text.