Due 09/13/13

** 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.

Due 09/20/13

** 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).

Due 09/27/13

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).

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.

Due 10/4/13

S_3={e,rho, rho^2,sigma,rho*sigma,rho^2*sigma}. Use this information to construct a Cayley table for S_3.

Due 10/11/13

which is necessary and sufficient for its image in Z_n to be a generator of Z_n.

is a subgroup of GL(n,R).

Due 10/18/13

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.

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.

Due 10/25/13

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.

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.