Math 4400, Spring 08, Sample problems Final Exam.

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1 Math 4400, Spring 08, Sample problems Final Exam. 1. Groups (1) (a) Let a be an element of a group G. Define the notions of exponent of a and period of a. (b) Suppose a has a finite period. Prove that the period divides the exponents of a. (c) Let H be a subgroup of G and suppose that some power a n of a belongs to H. Let k be the smallest positive power such that a k H. Prove that k divides any such n. (2) (a) Let G be a group with p elements, where p is a prime. Prove that G is cyclic. (b) Prove that a cyclic group is abelian. (3) Let G be a cyclic group. Let H be a subgroup of G. Prove that H is also cyclic, by describing a generator. (4) Let Z be the ring of integers. (a) Prove that any subgroup H of the additive group Z is a cyclic subgroup. (b) Prove that any ideal I of the ring Z is a principal ideal. (5) Let G be a group such that a 2 = e for all a G. Prove that G is abelian. (6) Let G be a group such that the function f : G G defined by f(x) = x 2 is a homomorphism. Prove that G is abelian. (7) Suppose G is generated by two elements x and y. Let a G. (a) Prove that if a commutes with x, it commutes with x 1. (b) Prove that if a commutes with x and y, it commutes with all elements of G. (8) Let G be a cyclic group of order n, with generator c. (a) Prove that c k is a generator of G iff gcd(k, n) = 1. (b) What are the generators of G =< c > if G has order 30. (c) In the same group, write down a cyclic subgroup of order 6. (d) Does this group contain a cyclic sugbroup of order 4? If so, write it down. If not, say why. (9) Let φ : G G be a homomorphism of groups. (a) Define ker(φ) and prove that it is a subgroup of G (b) Prove that ker(φ) is a normal subgroup of G. (c) Prove that Image(φ) is a subgroup of G. 1

2 2 (10) Prove the a homomorphsm φ : G G is injective iff ker(φ) equals e. (11) Let H, K be two subgroups of a group G. Let φ : H K G be defined as φ(h, k) = hk. Prove that φ is a one-to-one function iff H K = {e}. (12) Let H, K be two subgroups of a group G. Assume that H is a normal subgroup of G. Prove that HK is a subgroup of G. (13) Let H be a subgroup of G (a) Define the coset ah for any element a G. (b) Prove that a ah. (c) Prove that ah = H iff a H. (d) Prove that ah = bh iff a 1 b H. (14) Let G be a finite group and let H be a subgroup of G of index 2: this means that H has half the number of elements that G has. Prove that H is a normal subgroup in G. (15) State and prove the First Isomorphism Theorem for groups. (16) Let D 6 = {e, a, a 2, b, ab, a 2 b} be the dihedral group with 6 elements, and let Z/(3) = { 0, 1, 2} be the cyclic group with 3 elements, with + as the operation. Demonstrate that if ϕ : D 3 Z/(3) is a homomorphism, then ϕ cannot be onto. (17) Let G be a group and let H be a normal subgroup. Prove that G/H is abelian iff x, y G, x 1 y 1 xy H. (18) (a) Give the definition of the action of a group G on a set S. Explain the two different notations and indicate when one of the notations is more dangerous than useful. (b) Suppose G has a group action on a set S. Let s S. Define G s and O s. (c) Prove that G s is a subgroup of G. (d) Prove that two orbits in S are either disjoint or are the same. (e) Prove that the map φ : G/G s O s given by φ(gg s ) = gs, (or π g (s) if you prefer that notation) is first well-defined, and second a bijection. (19) (a) Explain why we can say that the number of conjugates of an element x in a group G is the index of the centralizer of x in G. Supply your definitions in your explanation. (b) Explain why we can say that the number of conjugates of a subgroup H of a group G is the index of the normalizer of H in G. Supply your definitions in your explanation. (c) Describe and explain the Class Equation of a group G.

3 (20) Use the Class Equation to prove that any p-group G has a nontrivial center. (21) Let G be an abelian group with 60 elements and consider the function f : G G defined by f(x) = x 3. (a) Verify that f is a homomorphism. (b) State Cauchy s Theorem. (c) Prove that f is not an isomorphism. (22) (a) Is the subset H = {e, [12], [34], [12][34]} of S 4 a subgroup? Verify this by writing the 4 4 multiplication table for H. (b) Is it a normal subgroup of S 4? Details? (23) Let V 4 be the subset of S 4 consisting of V 4 = {e, [12][34], [13][24], [14][23]}. (a) Verify that V 4 is a subgroup by computing the 4 4 multiplication table for V 4. (b) Is it a normal subgroup of S 4? Details? (24) Prove that if a group G contains exactly two subroups h and k of order 5, then hk = kh. (hint: the period of hkh 1 is 5. Why?) (25) Let H, K be two normal subgroups of G such that H K = {e}. Prove that for any elements h H, k K, hk = kh. (hint: study the commutator hkh 1 k 1.) 2. Rings (1) (a) Define a unit in a ring R (as opposed to the unit element e or 1). (b) Prove that R, the set of units in R, forms a group (under multiplication). (c) Prove that if a Z, then a Z/(n) is a unit iff gcd(a, n) = 1. (2) Let R be a commutative ring. Prove that R is a field iff the only ideals in R are the trivial ones ({0} and R itself). (3) Let f : R S be a ring homomorphism. Let I be a two-sided ideal in S. Let J = f 1 (I) = {x R f(x) I}. Prove that J is a two-sided ideal in R. (4) Let f : R S be a surjective ring homomorphism. Let M = ker(f). Define f : R/M S using the formula f(a + M) = f(a). (a) Prove that f is well-defined. (b) Prove that f is a homomorphism of rings. (c) Prove that f maps onto S. 3

4 4 (d) Prove that f is an isomorphism from R/M to S. (5) Let R be a commutative ring. (a) Define a prime ideal in R. (b) Define a maximal ideal in R. (c) Prove that any maximal ideal is a prime ideal. (d) Prove that the principal ideal (5) in Z is a maximal ideal in Z. (6) Let R be a commutative ring, P an ideal in R. Prove that P is a prime ideal iff R/P is an integral ring. (7) Let R be a commutative ring, M an ideal in R. Prove that M is a maximal ideal iff R/M is an field. (If you construct any auxilliary ideal in your proof, supply definitions and a reference to why this is an ideal.) (8) Let R be a ring of characteristic 5 and S a ring of characteristic 3. Show that if you claim there is a ring homomorphism f : R S, you get a contradiction. (9) Discuss: there is a ring homomorphism from the real numbers to the rational numbers. (10) In this problem, for each part, choose a ring that gives you the required example. In such a ring, find (a) a non-zero nilpotent element. (b) a divisor of zero. (c) an ideal which is not prime. (d) an ideal which is not maximal. (e) an ideal which is maximal. (f) a unit different from the unit element. 3. Last problems (1) Q is a subring of R (in fact a subfield). Let α R. Consider ev α : Q[t] R. It is customary to denote the image of this map by Q[α]. Since ev α is a ring homomomorphism, its image Q[α] is a subring of R. (a) Suppose α = 2 Q. Verify that the image of ev 2 is Q itself. (b) What is the kernel of ev 2? (You may need the Remainder Theorem.) (c) Use the last two parts and the First Isomorphism Theorem for rings to conclude that Q[t]/(t 2) = Q. (2) With the same terminology as in the last problem: Suppose α = 3 R. (a) Verify that the image of ev 3 is the set {a + b 3 a, b Q}.

5 (b) Argue that the kernel of ev 3 is the ideal (t2 3). (c) What does the First Isomorphism Theorem say now? (3) Use similar terminology for R viewed as a subfield of C, the complex numbers. Choose α to be the complex number i. (a) Prove that ev i : R[t] C is onto. (b) Argue that the kernel is the ideal (t 2 + 1). (c) Conclude that the First Isomorphism Theorem tells you that R[t]/(t 2 + 1) = C. (4) Let K[t] be the ring of polynomials over a field K. Let f(t) K[t]. (a) State and prove the Remainder Theorem in K[t]. (b) Use it to prove that α K is a root of a polynomial f(t) iff t α divides f(t). (c) Prove that if f(t) is a polynomial of degree n, then f can have at most n distinct roots in K. (5) Let K be a field. (a) State the Division (or Euclidean) Algorithm in K[t]. (b) Prove that if I is an ideal in K[t], then f I such that I = (f). (c) Suppose f K[t] is a cubic polynomial such that the ideal (f) is not a maximal ideal. Prove that there is a linear polynomial in K[t] which divides f(t). (Hence f has a root in K!) (6) Let p(t) be a polynomial in K[t] (K a field). (a) Define p(t) is an irreducible polynomial in K[t]. (b) Show that t is irreducible in R[t] but not in C[t]. (c) In which of the following fields K is t irreducible: K = Z 3, K = Z 5, K = Z 7, K = Z 11? (7) t 3 + t + 1 is an irreducible polynomial in Z 5 [t]. Hence by our theorems, Z 5 [t]/(t 3 +t+1) is a field. Use bar notation to denote the cosets (so t 2 + 2t stands for the coset of t 2 + 2t). (a) Find the multiplicative inverse of t in this field. (b) Find the multiplicative inverse of t 2 + 2t in this field. 5

Name: Solutions Final Exam

Name: Solutions Final Exam Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] All of