HW

Problems

Due Date

1

2

 1. Express elements in S_3 in cyclic notation and determine their orders. 

2. For S_5, list all possible cycle structures. For each cycle structure, give the number and find the order of elements with this cycle structure. Do a permutation and its inverse have the same cycle structure? why? How many conjugacy classes does S_5 have?

3. Show that an element x in a group is its own inverse if and only if x^2 = e. How many elements in S_5 are their own inverses?

4. Let S be a conjugacy class of the group G.

(a) Suppose x is in S. Show that S = {gxg^{-1}: g in G} = hSh^{-1} for all h in G.

(b) Show that S = {x} if and only if x commutes with all elements in G. Conclude that G is commutative if and only if all conjugacy classes are singleton.

(c) Show that two conjugacy classes of G are either identical or disjoint.

5. (a)Show that for s in S_n, sgn(s) = sgn(s^{-1}).

    (b) Show that for n > 1, S_n has as many even as odd permutations.

Jan 30

3

1.  #2.21 (ii), (iv)-(x).

2. Let H and K be two subgroups of the group G. Show that the intersection of H and K is also a subgroup of G.

3. Let G=<a> be a cyclic group of order 12. List all subgroups of G. For each group, find all generators.

4. Let G=<a> be a cyclic group of infinite order. Find all subgroups of G. For each subgroup, find all generators.

5. Let a be an element of order n in the group G. 

(a) Show that if a^k = e, then n divides k.

(b) Show that for any integer k > 1, the order of the cyclic group <a^k> is n/gcd(k, n).

6.#2.52 all except (i) and (ii).

7. Show that an index 2 subgroup H of the group G is normal in G.

Feb 6

4

1. Let H be a subgroup of the group G, and let g be any element of G. 

(a) Show that gHg^{-1} is a subgroup of G.

(b) Show that gHg^{-1} is isomorphic to H.

(c) Show that the intersection of gHg^{-1} over all elements g of G is the largest subgroup of H normal in G.

2. Show that the center Z(G) of the group G is an abelian subgroup.

3. Let <a> be a cyclic subgroup. Show that (a) given any integer k, the map f(x) = x^k is a homomorphism from <a> to itself; (b) f is an isomorphism of <a> if and only if a^k is a generator of <a>.

4. Show that groups A = <(1 2 3 4)> and B = {e, (12), (34), (12)(34)} are not isomorphic.

5. Define a new operation * on Z by x*y = x+y+2. HW1 #3 says that (Z, *) is a group. Is it isomorphic to (Z, +)?

6. Find the center of GL_2(Q).

7. #2.64

Feb 13

5

1. Let B be the set of upper triangular matrices in GL_2(Q), T be the set of diagonal matrices, and U be the set of matrices in B with diagonal entries 1.

(a) Show that B, T, U are subgroups of GL_2(Q).

(b) Show that U is normal in B, but not normal in  GL_2(Q).

(c) Show that B = TU.

(d) Show that the quotient group B/U is isomorphic to T.

2. Let G be a group of order 4. Show that either G is cyclic or G ={e, a, b, ab}, where a, b and ab all have order 2. Conclude that G is abelian.

3. Find all noncyclic order 4 subgroups of S_4. Which of these are normal in S_4? Give reasons.

4. Let G=(Z/mnZ, +), where m and n are coprime integers. Let

H = {h \in G : order h divides m} and K = {k \in G : order k divides n}.

(a) Show that the intersection of H and K is {0}.

(b) Show that H + K = G.

(c) Show that G/H is isomorphic to K and G/K is isomorphic to H.

(d) Show that H is isomorphic to Z/mZ and K isomorphic to Z/nZ.

 Feb 25

6

1. Let G be a cyclic group and H a subgroup of G. Show that G/H is cyclic.

2. Let G = Z/12Z and K = <4+12Z>.

  (a) Show that G/K is isomorphic to Z/4Z.

  (b) Find all subgroups H of G containing K.

3. #2.95.

4. Find all subgroups of (Z/p^2Z) x (Z/pZ), where p is a prime. List them according to their orders.

5. Classify all abelian groups of order 400.

6. Let G be an abelian group of order mn, where m and n are coprime integers. Let H = {g in G : g^m = e} and K = {g in G : g^n = e}.  Prove (a) and (b) below WITHOUT using the fundamental theorem of finite abelian groups.

(a) H and K are subgroups of G with intersection {e}.

(b) G = H x K. (Hint: Show G = HK then apply prop.. Let g be an element in G. Write the order of g as m'n', where m'|m and n'|n. Show that g^{m'} lies in K and g^{n'} lies in H, and manage to conclude g in HK from here.) 

Mar. 6

7

1. If a ring is  field, then its characteristic is either 0 or a prime.

2. #3.1.

3. (a) Show that Z[i] is an integral domain.

    (b) Show that Q[i] is a field.

    (c) Show that Q[i] is the fraction field of Z[i].

4. Show that the only ring homomorphism from Z to Z is the identity map, and the same holds from Q to Q.

5. #3.17

6. #3.29

 Mar 20

8

1. #3.41 (ii)-(ix)

2. (a) Determine the kernel of the ring homomorphism from Q[x] to Q[\sqrt {-5}]

sending g(x) to g(\sqrt {-5}).

    (b) Show that Q[\sqrt {-5}] is a field which is isomorphic to Q[x]/(x^2 + 5).

3. Show that (a) an ideal in Z is a prime ideal if and only if it is generated by a prime; (b) an ideal of Z is maximal if and only if it is a prime ideal not equal to 0 and Z.

4. Find (a) all units in Z/12Z; (b) all ideals of Z/12Z containing the ideal generated by 4+12Z.

5. Show that the ideal (2, 1+\sqrt{-5}) in Z[\sqrt{-5}] is not principal. (Hint: Consider the absolute value.)

6. Let R be a P.I.D. and I be an ideal of R. Show that R/I is also a P.I.D.

Mar 27

9

1. Show that a polynomial of degree n with coefficients in a field cannot have more than n roots counting multiplicity.

2. Find the polynomial f(x) of degree at most 2 with coefficients in Z/5Z satisfying f(1)=3, f(2)=0 and f(3)=2.

3. Let f(x) = x^3 -x^2 -x +1 and g(x) = x^3 + 4x^2 + x -6. 

Find gcd(f, g) and express it as a linear combination of f(x) and g(x) in (a) Q[x] and (b) Z/7Z[x].

4. Find lcm(f, g) for f(x) and g(x) in #3.

5. #3.56 all except (vii).

6. Let p be a prime number. 

(a) Factor x^{p-1} - 1 as a product of irreducible polynomials over Z/pZ. 

(b) Let f(x) and g(x) be two polynomials over Z/pZ. Show that f(a) = g(a) for all a in Z/pZ if and only if x^p - x divides f(x) - g(x).

April 10

10

1. Show that a prime element in a ring is irreducible.

2. Show that an Euclidean domain is a P.I.D. (Hint: Consult the proof that a polynomial ring over a field is a P.I.D.) 

3. Show that 1 + \sqrt {-5} is an irreducible element, but not a prime element, in the ring R = Z[\sqrt{-5}].

4. Show that there is a monic irreducible polynomial of degree 2 over the finite field Z/pZ for every prime p.

5. Show that x^2 + x + 1 is irreducible over Z/pZ if and only if p is congruent to 2 mod 3. (Hint. Consider (x-1)(x^2 + x + 1) to understand the roots of x^2 + x + 1.)

6. (a) Show that a polynomial of odd degree over the field R of real numbers has at least one root in R.

(b) Show that an irreducible polynomial over R has degree 1 or 2. Further, if it has degree 2, then its roots in C are not real, but complex conjugates of each other.

7. #3.82.

 April 17

11

1. #3.84

2. Suppose the field K is a degree n extension of the field F.

(a) Show that every element in K is algebraic over F.

(b) Show that the irreducible polynomial over F of an element of K has degree dividing n.

3. Suppose L, K and F are three fields such that L is a degree m extension of K and K is a degree n extension of F. Show that L is a degree mn extension of F.

4. Find the irreducible polynomial of (a) a = 3 - \sqrt 2 and (b) 

b= \sqrt 2 + \sqrt 3 over Q and Q(\sqrt 3) respectively. What are the degrees of the fields Q(a) and Q(b) over Q?  

5. (a) Let L be a finite extension of the field F, and let K be a subfield of L containing F. Show that the degree of K over F divides the degree of L over F.

(b) Suppose that the degree of L over F is a prime p. Show that L = F(a) for any a in L but not in F.

(c) Show that any field extension of F of prime degree p is isomorphic to  F[x]/(h(x)) for some irreducible polynomial h(x) over F of degree p. 

April 27

12

1. #3.92

2. Find the smallest extension of Z/pZ containing all roots of 

    (a) x^4 -1; (b) x^7 - 1.

13