HW Problems Due Date 1

1.  (a) Find the groups A and B of isometries of a square and a tetrahedron, resp. (b) Are the two groups

the same? If not, exhibit one element which lies in A but not in B, and conversely, in B but not in A.

2. #2.36. all except (vi), (vii), (ix);

3. Define a new binary operator * on the set of integers Z by x*y=x+y+2.  Show that (Z, *) is a group.

4. Let GL_2(R) be the set of 2x2 matrices with real entries. Show that under the matrix

multiplication GL_2(R) is a nonabelian group.

5. Classify all cyclic subgroups of (Z/8Z, +) and (S^1, x). Here S^1 is the set of complex numbers

with absolute value 1 and x is multiplication of complex numbers.

6. Show that each element in a group has a unique inverse.

Jan 21

 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= be a cyclic group of order 12. List all subgroups of G. For each group, find all generators. 4. Let G= 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 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 be a cyclic subgroup. Show that (a) given any integer k, the map f(x) = x^k is a homomorphism from to itself; (b) f is an isomorphism of if and only if a^k is a generator of . 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