Problem Set “Suggested Problems” Compilation, PSet 6-10 Math 350, Fall 2019

Problem Set “suggested problems” compilation, PSet 6-10 Math 350, Fall 2019 Below are all of the additional practice problems suggested on problem sets 6 to 10 (those covered on midterm 2). These are all useful for exam review. I will put at least one of these problems (perhaps with minor modifications) on the exam. 1. (3.1.3) Order of Elements in Sn . Prove Proposition 3.4. In other words, prove that the order of an element σ ∈ Sn is the least common multiple of the lengths of the cycles in its cycle decomposition. 2. (4.4.16) Find the number of conjugacy classes of S4 and the number of elements in each of these classes. 3. (5.1.1) Let G = Z/4Z × Z/3Z be the direct product of (Z/4Z, +) and (Z/3Z, +), and let H = h(2, 0)i be a subgroup of G. Find the right cosets of H in G. 4. (4.3.8) Let G = S4 , and let V = R4 . In Problem 7, we defined an action of G on V . √ √ (a) For this action, what is the stabilizer of (3, 2, 3, 2)? Find a familiar group that is isomorphic to this stabilizer. (b) For g ∈ G, let W (g) = {v ∈ V | g · v = v}. Prove that W (g) is a subspace of V . Find a basis for W (g) when g = (1 3) ∈ S4 . 5. (4.4.13) Let G = S3 and V = R3 . In√ Problem 7, we defined an action of G on V . (Also see Problem 4.) What is the orbit of (3, 2, 3) in this action? What about (4, 4, 4)? What are the possible orbit sizes for this action? 6. (5.2.2) If G is a noncyclic group of order 27, then for how many elements x of G do we have x9 = e? 7. (4.1.4) Let G be a subgroup of Sn . Hence every element of G is a permutation of [n] = {1, . . . , n}. Let ei be the element of Rn with a 1 in the ith coordinate and zeros in all other coordinates. The set B = {e1 , . . . , en } is the standard basis for Rn . Define an action of G on B by σ · ei = eσ(i) . Extend this action to an action of G on Rn as follows: If v ∈ Rn then, for some scalars α1 , . . ., αn , we have v = α1 e1 + · · · + αn en . For σ ∈ G, we define σ · v = α1 eσ(1) + α2 eσ(2) + · · · + αn eσ(n) . √ (a) Let n = 3, let G = S3 , and let v = ( 2, −8, 4) ∈ R3 . Find σ · v and τ · v, where σ = (1 2 3) and τ = (2 3). (b) Show that the above definition does indeed give an action of G on Rn . (c) Can you generalize the above action to an action of any subgroup of Sn on any n dimensional vector space with a designated basis? due never. page 1 of 3 Problem Set “suggested problems” compilation, PSet 6-10 Math 350, Fall 2019 8. (5.1.6 ) Let G be a group, and let H ≤ G with |G : H| = 2. (a) If K is a subgroup of G with at least one element not in H. Show that G = HK. (b) Is it possible to find y ∈ G such that yH 6= Hy? 9. (5.1.10) Let G be a group, and let H ≤ G. Recall Definition 4.24 of NG (H), the normalizer of H in G. Show that NG (H) = {x ∈ G | xHx−1 = H} = {x ∈ G | xH = Hx}. 10. (5.1.2) Let G = (Z, +) be the group of integers, and let H = (5Z, +) be the subgroup of G consisting of all multiples of 5. Describe the right cosets of H in G. 11. (5.2.5) Let D10 = ha, b | a5 = b2 = e, ba = a4 bi be the dihedral group of order 10. Assume x and y are two distinct elements of order two in D10 . Let H = hx, yi. What can you say about |H|? Can x and y commute? Give your reasons. 12. (10.1.1) Find all normal subgroups of D8 and of S3 . 13. (10.1.9) Find a group G, with subgroups H and K, such that H
K, K
G, but H not normal in G. 14. (10.2.4 ) If M and N are normal subgroups of a group G, show that M ∩ N is also a normal subgroup of G. 15. (10.3.3) Let G = Z/4Z × Z/6Z and let H be the subgroup of G generated by (2, 2). (a) What are the elements of H? (b) What are the elements of G/H? (c) Find a familiar group that is isomorphic to G/H. 16. (10.3.9 ) Let G be a group and let N
G. Assume that |G : N | = m. Let x ∈ G. Prove that xm ∈ N . 17. (11.3.5) Let D8 and S3 , as usual, be the dihedral group of order 8 and the symmetric group of degree 3 respectively. Assume φ : D8 → S3 is a homomorphism. What are the possibilities for |ker(φ)| and |Im(φ)|? For each possibility, give an explicit example. 18. (15.1.3 ) Let d √ be an integer (positive or negative) not divisible by a square√of a prime, and √ √ Z[ d] = {a + b d | a, b√∈ Z}. Let N : Z[ d] → Z be defined by N (a + b d) = a2 − db2 . Prove that, for x, y ∈ Z[ d], we have N (xy) = N (x)N (y). due never. page 2 of 3 Problem Set “suggested problems” compilation, PSet 6-10 Math 350, Fall 2019 19. (15.1.5) Show√that, without using ±1 as one of the factors, neither 3 nor 2 + factored in Z[ 5i]. √ 5i can be 20. (15.2.6) Is Z/2Z × Z/2Z a field? Is Z/4Z a field? Can you find a field with four elements? If so, give its addition and multiplication tables explicitly.   a b 21. (15.2.7) Let F2 = (Z/2Z, +, ·) and define E = { | a, b ∈ F2 }. How many elements b a+b does E have? With the usual matrix addition and multiplication, is E a field? 22. (15.2.11) Let X be a non-empty set, and recall (Definition 2.20) that 2X is the set of all subsets of X, and for A and B subsets of X, their symmetric difference is denoted by 4 and is defined by A4B = (A − B) ∪ (B − A). Show that (2X , 4, ∩) is a commutative ring with identity. Is it an integral domain? 23. (15.2.13) Find the group of units of Z/5Z, Z/6Z, Z/12Z, and Z/24Z. 24. (16.1.1) If D is an integral domain and R a subring of D with at least two elements, then is R necessarily an integral domain? Either prove that it is, or give an example where it is not. 25. (16.1.4) Proof of Theorem 16.12d and 16.12e. Let R and S be rings, and φ : R → S a ring homomorphism. Let R0 and S 0 be subrings, respectively, of R and S. Prove that φ(R0 ) and φ−1 (S 0 ) are subrings, respectively, of S and R. 26. (16.1.7) Let R be a ring with identity. How many ring homomorphisms φ : Z → R are there with φ(1) = 1R ? √ √ 27. (16.1.10) Let R = Q[ 2] and S = Q[ 3]. Show that the only ring homomorphism from R to S is the trivial one. In particular, conclude that R and S are not isomorphic rings. In other words, assume f : R → S is a ring homomorphism. Show that f (r) = 0 for all r ∈ R. 28. (16.1.20 ) Let R be a ring with identity, and let J be an ideal of R. Assume that J contains a unit of R. Prove that J = R. due never. page 3 of 3