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 (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.
(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 (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}.
(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 2 + 1 is irreducible in R[t] but not in C[t]. (c) In which of the following fields K is t 2 + 1 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