Abstract Algebra
Homework Assignment
Week 1
Due 09/14/12

 

Problem 1:  Do problem 1, Exercises 1.1 on page 28.

Problem 2:  Show that strong induction implies the well ordering axiom.
 

Problem 3:  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 4:  .Show that for nonzero integers m and n,  gcd(m,n) is the largest natural number dividing both m and n.
 

Problem 5:  Suppose that m and n are relatively prime integers and that m divides nx for some x. Show that m divides x.
 


Abstract Algebra
Homework Assignment
Week 2
Due 09/21/12

 

Problem 1: Suppose that m and n are relatively prime integers and that m|x and n|x.  Show that mn|x.

Problem 2: If a=b(mod n), and m|n, show that a=b(mod m).

Problem 3:  Compute gcd(32242,42) and express this gcd as a linear combination of the two numbers.

Problem 4:  Let p be a prime.  Show that if x^2=1 in Z_p, then either x=1 or x=-1.  Use this result to prove Wilson's Theorem: (p-1)!=-1(mod p).

Problem 5:  Suppose n is not equal to zero, Show that in Z_n, every nonzero element is either invertible or a zero divisor. (A zero divisor is an nonzero element a such that ab=0 for some nonzero b).
 

Problem 6:  Compute 4^237 (mod 12).

Problem 7:  If a is relatively prime to n, then there are integers x and y such that xa+yn=1.  We also know an algorithm for computing such an x and y.  Use this idea to give an algorithm to
                    compute the inverse of  a (mod n).  

Problem 8:  Show that if a and n are relatively prime, then ax=b (mod n) has a solution.  Give an algorithm for constructing such a solution and use it to solve 8x=20 mod (81).


Abstract Algebra
Homework Assignment
Week 3
Due 09/28/12

Problem 1:  Prove that the "Freshman Exponentiation" rule holds in Z_p, where p is a prime (a+b)^p=a^p+b^p.

Problem 2:  Show  that a^5 is congruent  to either 0, 1 or -1 (mod 11) for every integer a.

Problem 3:  If ab=ac in Z_n and a is invertible, show that b=c.

Problem 4: A Carmichael number is a composite number which satisfies the conclusion of Fermat's Little Theorem. In other words, n is a Carmichael number if
                   whenever a and n are relatively prime  a^n=a mod n.   Show that 561 is a Carmichael number.  (Hint: see exercise 37 on page 51).

Problem 5: In S_3, let sigma=(1,2) and rho=(1,2,3), written in disjoint cycle notation and e be the identity element.  Show that sigma^2=e, rho^3=e and that sigma*rho=rho^2*sigma.
                   Furthermore, show that the 6 elements of S_3 are precisely the set {e,rho, rho^2,sigma,rho*sigma,rho^2*sigma}.  Use cycle notation in your arguments.


 


Abstract Algebra
Homework Assignment
Week 4
Due 10/5/12

Problem 1:  Show that the set of integers Z, with the operation  a*b=a+b-1 is an abelian group. 

Problem 2:  In the symmetric group S_3, let sigma=(1,2), and  rho=(1,2,3). In last week's homework you showed that sigma*rho=rho^2*sigma, and that
                    S_3={e,rho, rho^2,sigma,rho*sigma,rho^2*sigma}.  Use this information to construct a Cayley table for S_3. 

Problem 3:  Suppose that G is abelian, and H={g in G | g^2=e}. Show that H is a subgroup of G.  Give an example of a nonabelian group where this does not hold.

Problem 4: Show that if H is a subgroup of G and K is a subgroup of H, then K is a subgroup of G.

Problem 5: Show that if G is a group, and H is a nonempty subset of G, then H is a subgroup of G provided that whenever x and y are in H, then x*y^(-1) is in H.


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

Problem 1:  For the symmetric group S_3, find all subgroups.  For each subgroup, list the elements.  Which of these subgroups is cyclic?

Problem 2:  If G=<g>, then g is said to be a generator of the group G.  Find all generators of the groups Z_4, Z_6 and Z_7.  Formulate a condition on an element  k  in Z,
                    which is necessary and sufficient for its image in Z_n to be a generator of Z_n.

Problem 3:  Find an element of largest possible order in S_5

Problem 4: Draw the subgroup lattice diagrams for Z_12 and Z_18 and S_3.

Problem 5: Give an example of a group G, and an abelian subgroup H of G, such that H does not lie in the center Z(G) of G.

Problem 6: If g is an element of a group G, then the centralizer C(g) of g is defined by C(g)={x in G| xg=gx}.  Show that the centralizer of g is a subgroup of G.

Problem  7: Recall that GL(n,R)={A is an nxn real matrix| A is invertible} is called the general linear group.  Show that the special linear group SL(n,R)={A in GL(n,R)| det(A)=1}
                    is a subgroup of GL(n,R).

Problem 8: Show that Z_12^*={a in Z_12|a is invertible} is not a cyclic group.



Abstract Algebra
Homework Assignment
Week 6
Due 10/19/12

Problem 1:  If k is an element of Z, show there is a unique group morphism  phi_k : Z -> Z from  the integers to the integers such that phi_k(1)=k.  Next show that if phi : Z -> Z is any
                    group morphism, then phi=phi_k for some k.  Finally,  determine for which k the morphism phi_k is an isomorphism, and for which k the morphism phi_k is injective.

Problem 2:  Show that every infinite cyclic group is isomorphic to the integers.

Problem 3:  If g, h are elements of a group G show that the subgroups <gh> and <hg> are isomorphic.

Problem 4:  Show that the group of rational numbers under addition is not isomorphic to the integers.

Problem 5:  A bijective homomorphism of groups f: G -> G is called an automorphism.  Show that the set Aut(G) of automorphisms of G is a group.  Explain why it is a subgroup
                     of the group S_G of all permutations of the set G.

Problem 6: The group O(2) of all orthogonal 2x2 matrices is generated by elements rho_theta,  where rho_theta is the counterclockwise rotation by the angle theta, and the element sigma, which is the       
                    reflection over the x axis.  Find matrices representing rho_theta and sigma, and show the following facts. 
                    1) (rho_theta)^{-1} = rho_{-theta}.
                    2) sigma*rho_theta=(rho_theta)^{-1}*sigma.
                    Notice that these relations are very similar to the ones in the dihedral group.  Thus the group O(2) is a type of infinite dihedral group.


Abstract Algebra
Homework Assignment
Week 7
Due 10/26/12


Problem 1:  If G is a group and g is in G, then define c_g: G -> G by c_g(x)=gxg^{-1}.  This is called conjugation by g.  Show that the set Int(G)={c_g| g in G} is a subgroup of Aut(G).
                    Int(G) is called the group of inner automorphisms of G.  Next show that if phi is an automorphism of G, then phi c_g phi^{-1} =c_{phi(g)}.  Why does this show that Int(G) is
                    a normal subgroup of G?  The group Out(G) = Aut(G)/Int(G) is called the group of outer automorphisms of G.

Problem 2:  Define a map G -> Int(G) by g -> c_g.  Show that this map is a morphism of groups.

Problem 3:  The dihedral group D_n is given by generators rho and sigma, with the relations rho^n=e, sigma^2=e and sigma*rho=rho^{-1}*sigma.  Show that these relations imply that
                     every element in D_n is of the form rho^k or rho^k*sigma, where 0<=k<n.  This means that there are exactly 2n elements in the dihedral group.  Make a Cayley table for the group
                     D_3.

Problem 4:   Show that the center Z(G) of any group is normal. Find the subgroup Z(D_6). Show that D_6/Z(D_6) is isomorphic to D_3. 

Problem 5: Find the commutator subgroup G' of the permutation group G=S_3.  Then determine which group G/G' is isomorphic to.

Problem 6: Suppose that H<=G is a subgroup and that |G: H| =2.  Show that H is a normal subgroup of G.