1. Group Theory Permutations.


 Johnathan Blake
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1 1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7 be of maximal order. What is π? Is π an even permutation? If ρ = π, is ρ conjugate to π? Problem 1.3. Let n 3, and let σ S n be an (n 1)cycle. Show that C Sn (σ) = σ. Problem 1.4. Let σ = (123)(456) S 6. Find the size of the conjugacy class of σ, and the order of the centralizer of σ in S 6. Problem 1.5. S n has an element of order 2(n 2) whenever n > 1 is odd Sylow theorems. Problem 1.6. How many elements of order 7 are there in a simple group of order 168? Problem 1.7. A group of order 48 must have a normal subgroup of order 8 or 16. Problem 1.8. Let G be a group of order p a b where (p, b) = 1, b > 1 and p a b > b!. Then G is not simple. In particular, there is no simple group of order 36. Problem 1.9. Let G be a finite group, H G, and let P be a Sylow psubgroup of H. Assume that N G (P ) H and conclude that P is a Sylow psubgroup of G. Problem A group of order 340 has a normal cyclic subgroup of order 85 and an abelian subgroup of order 4. Problem A group of order 108 has a normal subgroup of order 9 or 27. Problem Let p, q be distinct primes. Then any group of order p 2 q either has a normal Sylow psubgroup or it has a normal Sylow qsubgroup. Problem Proof Frattini argument, i.e.: Let G be a finite group, H G, P a Sylow psubgroup of H. Then G = N G (P )H. Problem Let N be a normal subgroup of G that contains a Sylow psubgroup of G. Then the number of Sylow psubgroups of N is the same as that of G. Problem Let G be a group of order 105, then G has a normal Sylow 5subgroup and a normal Sylow 7subgroup Actions. Problem Let G be a finite simple group containing an element of order 21. Show that every proper subgroup of G is of index at least 10. Problem Let G be a simple group with G > 60. Show that G has no subgroups of index less or equal to 5. Problem Let G be a group acting on set S, H G, such that the inherited action of H on S is transitive. Show that for every t S we have G = HG t, where G t is the stabilizer of t. Problem Let G act on Σ transitively, α, β Σ, α β. Show that G α G β G. Problem Show that S 6 has no simple subgroup of order 180 (index 4). Then use this fact to prove that no simple group of order 180 = exists. Problem Prove that there are no simple groups of order 1452 =
2 Solvability and nilpotency. Problem It is true (and you can take it for granted) that a group of order 12 is either isomorphic to A 4 or contains an element of order 6. Show that a group of order 12p is solvable whenever p > 11 is a prime. Problem Show that every group of order is solvable. Problem Let G be a group of order 780 = that is not solvable. What are the composition factors of G? You can take for granted that the only nonabelian simple group of order at most 60 is A Classical problems. Problem Show that the additive group Q of the rational numbers is not cyclic. Show that every finitely generated subgroup of Q is cyclic Other structural problems. Problem Let G be a simple group of order at least 3. Then Z(Aut(G)) = 1 if and only if G is not abelian. Problem Let M be a maximal subgroup of a finite group G. If M G, show that [G : M] is prime. Problem Let G be a nonabelian group of order p 3. Show that Z(G) = G is a subgroup of order p, and that G/Z(G) = Z p Z p. Problem A finite group whose only automorphism is the identity map must have order at most two. Problem Let G be a finite group, N G, H G, ([G : N], H ) = 1. Then H N. Problem Find all finite groups that have exactly two conjugacy classes. Problem Find all abelian groups of order 108 (up to isomorphism) Ideals. 2. Rings Problem 2.1. Let R be a commutative ring with 1. Show that the sum of any two principal ideals of R is principal if and only if every finitely generated ideal of R is principal. Problem 2.2. Let R 1, R 2 be commutative rings with identities and let R = R 1 R 2. Show that every ideal I of R is of the form I = I 1 I 2 with I i an ideal of R i, for i = 1, 2. Problem 2.3. Let R be a commutative ring with unity, S a subset of R closed under multiplication, and P a maximal element of the set {I; I is an ideal of R such that I S = }. Show that P is a prime ideal of R. Problem 2.4. Let R be a commutative ring with 1. Show that if M is a maximal ideal of R then M is a prime ideal of R. Problem 2.5. Let R be a commutative ring that is not a field, and let P 0 be a maximal ideal of R. Show that P [x] is a prime ideal of R[x] but P [x] is not a maximal ideal of R[x].
3 Problem 2.6. Give an example of a commutative ring R with prime ideal P 0 that is not maximal. Problem 2.7. Let R be a commutative ring with 1 0 such that every ideal of R is prime. Then R is a field. Problem 2.8. Let R be an integral domain with 1 0 where IJ = I J for all ideals I, J of R. Then R is a field. Problem 2.9. Let R be a commutative ring with 1. We say that R satisfies the ACC if whenever I 1 I 2 I 3 is a chain of ideals of R, then there exists an integer N such that I k = I N for every k N. Prove that R satisfies the ACC if and only if every ideal of R is finitely generated. Problem Let F be a field and F [x, y] the ring of polynomials in variables x, y. Show that the ideal of F [x, y] generated by {x, y} is not principal. Problem Let R be a commutative ring with 1 such that it has precisely three ideals, say {0}, I and R. Then (i) every a R \ I is a unit, (ii) if a, b I then ab = 0. Problem Let R be the ring of 2 2 matrices with coefficients in some field F. Show that R is simple, i.e., it has no proper nonzero ideals. Problem Let C be a chain of prime ideals of a commutative ring R with 1. (Do not assume that the chain has a minimal or maximal element, nor that it is finite, countable, etc.) Show that C C C and C C C are prime ideals of R. Problem Let p be a prime and R the ring of all 2 2 matrices of the form ( a b pb a), where a, b Z. Then R is isomorphic to Z[ p]. Problem Let R be a nonzero commutative ring with 1. Show that if I is an ideal of R such that 1 + a is a unit for every a I, then I is contained in every maximal ideal of R. Problem Let R be a commutative ring with 1, and let P be a prime ideal of R. Show that there is a prime ideal P 0 P that contains no prime ideals properly. Problem A commutative ring with 1 is local if it has a unique maximal ideal. Let R be a commutative ring with 1. Show that R is local iff the set of nonunits in R is an ideal Irreducible and prime elements. Problem Prove that in an integral domain R every prime element is irreducible. Problem Construct the following: (i) An integral domain with an irreducible element that is not prime. (ii) A commutative ring with a prime element that is not irreducible. Problem Let D be a UFD and F the field of fractions of D. irreducible in D. Prove that there is no x F such that x 2 = d. 3 Let d D be Problem Let D be an integral domain an c an irreducible element of D. Show that the ideal (x, c) is not a principal ideal in D[x]. Then show that the same conclusion cannot be reached if the assumption that c is irreducible is dropped.
4 Polynomials and quotient rings. Problem Find all values of a Z 5 such that the quotient ring R = Z 5 [x]/(x 3 + 2x 2 + ax + 3) is a field. Problem Show that the ring R = Z 2 [x]/(x 2 + 1) has 4 elements and that it is not isomorphic to Z 4 nor Z 2 Z UFDs. Problem Let D = Z[ 21] = {m + n 21; m, n Z}, and let F = Q( 21) be the field of fractions of D. Show that: (i) x 2 x 5 is irreducible in D[x] but not in F [x], (ii) D is not a UFD, (iii) D[x] is not a UFD. Problem Let Z[1/2] = {a/2 n ; ; a, n Z, n 0}. Show that Z[x]/(2x 1) = Z[1/2]. Then find an ideal I of Z[x] such that (2x 1) I Z[x] Examples. The following problem would not appear in its entirety as a single question on the qualifier. Problem Give an example for each of the following: (i) An infinite noncommutative ring with nonzero characteristic. (ii) An integral domain which is not a unique factorization domain. (iii) A noncommutative domain that is not a division ring. (iv) A nonzero prime ideal of a commutative ring that is not a maximal ideal. (v) A commutative ring with a sequence of prime ideals {P n } n=1 such that P i P i+1. (vi) A commutative ring that has exactly one maximal ideal and is not a field. (vii) A finite noncommutative ring. (viii) A noncommutative ring with exactly two maximal ideals. (ix) An infinite noncommutative ring with only finitely many ideals. (x) A unique factorization domain that is not a principal ideal domain. (xi) An infinite domain of nonzero characteristic Classical problems. Problem Let R be the ring of 2 2 real matrices of the form ( a b b a). Show that R is isomorphic to C. Proof. Let R be a nonzero ring with 1. Show that every proper ideal of R is contained in a maximal ideal. Problem Let R = Z/60Z. How many units does R have? How many ideals does R have? Problem Let R be a commutative ring with identity. Show that the nilpotent elements form an ideal of R Finite Fields. 3. Fields Problem 3.1. Construct a field of order 8. Problem 3.2. Construct a field of order 9.
5 Problem 3.3. Let α be a root of x in an extension of Z 3. Let F = Z 3 (α) and f(x) = x Z 3 [x]. Then: (i) Show that f splits in K. (ii) Find a generator β of K. (iii) Express the roots of f in terms of β Extensions. Problem 3.4. Let D be an integral domain with subring F that happens to be a field. Assume that D is algebraic over F. Then D is a field. Problem 3.5. Show that every finite extension of fields is algebraic. Problem 3.6. Let F E. Show that the algebraic closure of F in E defined by A = {u E; u is algebraic over F } is a field. Problem 3.7. Give an example of an algebraic extension that is not finite. Problem 3.8. Find the minimal polynomial of u = over Q Splitting Fields. Problem 3.9. Let E be a spitting field of f(x) F [x] over F, where deg f = n. Show that E : F n!. Problem Find a splitting field E of f(x) = x 4 2x 2 3 over Q. Assume that E is another splitting field of f over Q. What is E : Q? Problem Let E be a splitting field of f(x) = x 3 5 over Q. What is E : Q? Problem Let E be a splitting field of f F [x] over F, where deg f = 2. Show that F E is a simple extension. Problem Let A be the splitting field of f(x) = x 2 2 over Q, and B the splitting field of g(x) = x 2 2x 1 over Q. Show that A = B Theoretical problems. 4. Galois Theory Problem 4.1. Define: Galois group, Galois extension, Fixed field. Problem 4.2. Show that every finite extension of a finite field is a Galois extension Computational problems. Problem 4.3. In each case show that F E is Galois, determine the Galois group (isomorphism type and elements), find the lattice of intermediate fields, and describe the intermediate fields. (i) E = Q(u), u = e 2pi/5, F = Q, (ii) E = Q(i, 3), F = Q, (iii) E = Z 2 (u), where u is a root of x 4 + x + 1, F = Z 2. Problem 4.4. Determine the Galois group of x 4 1 and x 12 1 over Z 2. 5
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