Decide whether each of the statements below is true or false. Circle your choice.

2. (7 points) Find god(216,315) and express it in the form 216n +315m for integers n, m. ged(216, 315) =

3. (14 points) Prove that (Z,+), the integers with addition, forms a group (you may assume "standard" facts about addition of integers). 4. (14 points) Draw the subgroup lattice of Z100-

5. (14 points) Suppose that : G→ H is a group homomorphism, K <H is a subgroup, and ¹(K) = {g €G | (g) € K}. Prove that -¹(K) is a subgroup of G. 6. (14 points) Suppose A, B4G are normal subgroups such that AnB = {e}. Prove that if a EA and be B, then ab = ba. IMPORTANT: You should not simply quote the Proposition from class stating that AB Ax B. Instead, you should supply the first part of that proof of that proposition, stating that the elements of A commute with the elements of B.

7. Let o = (15)(26 9 7 8) (34), 7= (1 3 5)(29) € Sg and G=(0,7) < Sg. (a) (7 points) Find the composition a OT, expressed in disjoint cycle notation. JOT= (b) (7 points) Is a even or odd? Circle the correct answer. EVEN ODD (c) (7 points) Compute the order of a. |o|= (d) (7 points) Considering the action of G on {1,2,3,4,5,6,7,8,9} coming from the inclusion into S9, list the elements of the orbit of G. G.1= (e) (7 points) With the action of G in part (d), we have stabc(1) has 360 elements (you do not need to prove this). What is the order of G? Hint: Use your answer to part (d), even if you couldn't find G.1. |G|=

8. Let G be a nonabelian group of order 253 =23-11, let P <G be a Sylow 23-subgroup and Q<G a Sylow 11-subgroup. (a) (7 points) What are the orders of P and Q? |P| = |2|= (b) (14 points) How many distinct conjugates of P and Q are there in G? Justify your answer (no credit without justification). 123 = the number of conjugates of P 711 = the number of conjugates of Q= (c) (7 points) Prove that G is isomorphic to a semidirect product P x Q.

9. Let p(x) = 1+2+1²+1³ +r¹ € Q[r]. This polynomial is irreducible (you do not need to prove this). Consider the field obtained as the quotient K = Q[r]/I, where I = {fp|ƒ €Q[r]}. Let a = 1 + 1 € K, which is a root of p(x) in K, and recall that every element of K can be expressed in the form a+ba+co²+do²³, with a, b, c, d e Q. (a) (8 points) What is the degree of the extension, [K: Q? [K: Q = (b) (8 points) Compute the product of 1+a and 2-30³, and express it in the form a +bx+ co²+ da, with a, b, c, d € Q. (1 + a)(2-30³) = (c) (8 points) Since K is a field, and a #0, it follows that a is a unit. Find the multiplicative inverse ¹ and express it in the form a+ba+ca²+da³, with a, b, c, d e Q.

Exercise 3.3.4 Prove that if G is an abelian group, then every subgroup of G is normal.

Exercise 3.7.8 For every n23, construct a nonabelian group of order ny(n), where y(n) = Z is the Euler phi function of n. Hint: Use Exercise 3.7.7.

Exercise 4.1.1 Prove Proposition 4.1.5: Suppose Gx X → X is an action. Then for all z € X, stabĠ(z) is a subgroup of G and the kernel K of the action is the intersection of all stabilizers

Exercise 4.3.1 Verify the claims in the proof of Theorem 4.3.2. Specifically, prove that (i) a - a (91,..., 9p) = (ga+1,..., ga+p) defines an action of Z, on GP and (ii) X = {(9₁,..., 9p) EGP | 91929p = e} is invariant by Zp and hence the Z, action on GP restricts to an action on X. Hint: for the last part, you need to show that for all a € Zp, 91 9p = e if and only if ga+1 ga+p = e.

Exercise 4.3.4 Let p≥ 2 be a prime and prove that any group of order p² is isomorphic to either Zp² or Zpx Zp. Hint: If there exists an element of order p², then GZp², so assume that this is not the case and deduce that every nonidentity element of G has order p. Then prove that for any two elements g, h E G, either (g) = (h) or (g) n(h) = {e}. Finally, prove that there are two nonidentity elements g, h E G so that (g) n(h) = {e}, and apply Proposition 3.7.1.

Exercise 3 Let G be a group of order 52-72-19-23275. (a) Prove that G contains exactly one subgroup of order 49. Prove furthermore that if N <G with |N| then N is normal. (b) Prove that G/N is isomorphic to either Z19 × Z25 or Z19 × Z5 X Z Suggestion: Similar to the Exercise 2c, exept apply Proposition 3.7.1 instead of 3.7.7. 49 (c) Let Ps and P19 be Sylow 5- and 19-subgroups of G, respectively. Prove that NP5 and NP19 are both subgroups of G and that NPN X P5, and NP19 N x P19-

Exercise 4 Let G be a group of order 2³ - 7 = 56. Prove that G contains at least one normal subgroup. Suggestion: Prove that G contains either a normal Sylow 2-subgroup or a normal Sylow 7-subgroup. Do this by assuming otherwise and counting elements. Be careful with intersections when doing your count.

3.1.1 Describe the following sets in roster notation: that is, list their elements.

3.2.11 For the following proof sketch, determine the result that is being proved, and then turn the sketch into a formal proof.

Exercise 1 Let Z[i] = {a+bi | a,b€ Z}, and define the operation + in Z[i] as ordinary addition (as in C). Note that Z= { a+0i | a € Z } is naturally a subset of Z[i]. (a) Prove that (Z, +) is a normal subgroup of (Z[i], +). (b) Prove that the quotient group Z[i]/Z (with operation +) is an abelian group. (c) Let (5) denote the set (5) = { 5a +5bi | a,b € Z}. Prove that (5) is a normal subgroup of Z[i]. (d) Prove that Z[i]/(5) is an abelian group (with operation +).

Exercise 2 Over the course of the parts of this exercise you will show that multiplication of cosets in Z[i]/Z is not well-defined. (a) Let a, a, b, VEZ. Prove that a +i and a' + i represent the same coset in Z[i]/Z; that is, prove that (a + i) + Z= (a' + i) + Z (b) Prove that band & represent the same coset in Z[i]/Z; that is, prove that b+z=b'+Z (c) Determine the conditions under which (a +i)b and (a' + i)b' represent the same coset in Z[i]/Z. That is, determine what additionally must be assumed in order to ensure that (a + i)b+Z= (a' + i)b' +Z. Be sure to prove your condition is the correct one. (d) Find specific values of a, a, b, b' € Z for which (a+i)b+Z‡ (a¹ + i)b +Z.

Exercise 3.6.4 Suppose that G₁, G₂ are groups, N₁ ◄G₁, N₂ ◄ G₂ are normal subgroups. Prove that N₁ × N₂ ◄ G₁ x G₂, and (G₁ × G₂)/(N₁ × N₂) ≈G₁/N₁ × G₂/N₂. Hint: Theorem 3.6.5.

Exercise 3.7.2 Suppose G is a finite group, H, KG are normal subgroups, ged(|H|, |K|) = 1, and |G| = |H||K|. Prove that G H x K.