Homework

Homework is due each Tuesday at the beginning of class. Late homework will not be accepted!

Homework #10 (Due May 16)
4.2 (page 141): 8(a,b,c)
4.3 (page 147): 6
4.4 (page 152): 4
4) Describe all different colorings of the faces of a regular tetrahedron into 2 colors. Use Burnside's theorem to prove that these are all of them.
Bonus: Let G be a non-Abelian group of order p³ for p prime. Find the number and the sizes of the conjugacy classes in G. (Hint: Prove that the center has exactly p elements.)

Homework #9 (Due May 4)
3.4 (page 129): 8, 17(a), 19

Practice problems for Midterm 2

Homework #8 (Due April 18)
Quiz April 18: Know every definition in sections 2.1--2.5 by heart. For each definition you should be able to give examples that do satisfy the definition and examples that do not. There will be 3 questions on the quiz for 30 minutes.
2.4 (page 100): 3, 18, 22, 23(b)
2.5 (page 106): 14
Bonus: Is it true that if G/Z(G) is Abelian then so is G?

Homework #7 (Due April 11)
2.2 (page 87): 32, 36, 45, 48
2.3 (page 93): 8, 16, 24

Homework #6 (Due April 4)
2.1 (page 78): 2, 4, 19, 21, 32
2.2 (page 86): 7, 10, 16
Bonus: Is it true that if G is a group of order 12 then it has a subgroup of order 6?

Homework #5 (Due March 28)
1.4 (page 71): 10, 24, 32, 35, 40, 42
Bonus: Describe all possible rotations of the regular tetrahedron. (Don't forget rotations about the line through the midpoints of a pair of non-adjacent edges!) Each rotation corresponds to a permutation of the vertices of the tetrahedon. Determine the group of permutations whose elements correspond to the rotations of the tetrahedon.

Homework #4 (Due March 7)
Quiz March 7: Know every definition in sections 1.1--1.3 by heart (except for the centralizer). For each definition you should be able to give examples that do satisfy the definition and examples that do not. There will be 3 questions on the quiz for 30 minutes.
1.2 (page 55): 26 (b)
1.3 (page 60): 1(e), 6, 9, 17, 23 (c)
Bonus: Is it true that if any proper subgroup of G is cyclic then G itself is cyclic?

Homework #3 (Due February 28)
1.1 (page 48): 3, 9, 18, 22, 23
1.2 (page 54): 5, 18, 28
Bonus: Is it true that for any group G the set of all its elements of finite order forms a subgroup of G?

Homework #2 (Due February 16)
1) Prove Proposition 0.3.29 (page 19).
2) Let &phi(a) denote the number of positive integers less than a and relatively prime to a (Euler's function). We have seen in class that &phi(p) = p - 1, where p is prime.
   a) Prove that &phi(pq) = &phi(p)&phi(q), where p and q are distinct primes. (Hint: How many numbers less than pq are divisible by p? By q? By both?)
   b) Prove that &phi(p^k) = p^k - p^k-1 for any natural k.
   c) Prove that &phi(p^kq^l) = &phi(p^k)&phi(q^l) for any natural k, l.
   d) Compute &phi(6125).
3) Find all integers x that satisfy 17x = 2 mod 48.
4) In this problem we discuss the Chinese Remainder Theorem.
   a) Find an integer x which satisfies both congruences simultaneously:
      x = 0 mod 49
      x = 1 mod 33.
   b) In fact, there are infinitely many such x. Can you describe all of them?
   c) Let m and n be two relatively prime numbers. Show that the system
      x = a mod m
      x = b mod n.
   always has a solution for any a, b.
   d) Show that if x and x' are two solutions to the system then mn divides x - x'. Conclude that the system is equivalent to a single congruence
      x = c mod mn, for some c.
5) Write down all invertible 2 by 2 matrices with entries in Z₂. Explain why these are all of them.
6) Find all solutions to x³=8i.

Homework #1 (Due February 7)
0) Read Writing Proofs (by Professor Christopher E. Heil at Georgia Tech)
1) Prove that:
   a) if a divides b and b divides c then a divides c.
   b) if a divides b and c divides d then ac divides bd.
2) Is it true that:
   a) Given any 100 integers, one can choose 2 of them such that their difference is divisible by 99?
   b) (Bonus) Given any 100 integers, one can choose some of them (possibly all) such that their sum is divisible by 100?
3) Prove that the product of any 3 consecutive integers is divisible by 6.
4) (Bonus) Is it true that the product of any n consecutive numbers is divisible by n! (n factorial)?
5) Compute the gcd of 5586 and 364 using the Euclidean algorithm. Write the gcd as a linear combination of 5586 and 364 with integer coefficients.