Abstract Algebra Theory and Applications - Computer Science ...

HINTS AND SOLUTIONS 395Chapter 5. Cosets and Lagrange’s Theorem1. The order of g and the order h must both divide the order of G. The smallestnumber that 5 and 7 both divide is lcm(5, 7) = 35.2. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.3. False.4. False.5. (a) H = {0, 8, 16} 4 + H = {4, 12, 20}1 + H = {1, 9, 17} 5 + H = {5, 13, 21}2 + H = {2, 10, 18} 6 + H = {6, 14, 22}3 + H = {3, 11, 19} 7 + H = {7, 15, 23}.(c) 3Z = {. . . , −3, 0, 3, 6, . . .}1 + 3Z = {. . . , −2, 1, 4, 7, . . .}2 + 3Z = {. . . , −1, 2, 5, 8, . . .}.7. 4 φ(15) ≡ 4 8 ≡ 1 (mod 15).12. Let g 1 ∈ gH. Then there exists an h ∈ H such that g 1 = gh = ghg −1 g ⇒g 1 ∈ Hg ⇒ gH ⊂ Hg. Similarly, Hg ⊂ gH. Therefore, gH = Hg.17. If a /∈ H, then a −1 /∈ H ⇒ a −1 ∈ aH = a −1 H = bH ⇒ there exist h 1 , h 2 ∈ Hsuch that a −1 h 1 = bh 2 ⇒ ab = h 1 h −12 ∈ H.Chapter 6. Introduction to Cryptography1. LAORYHAPDWK.3. Hint: Q = E, F = X, A = R.4. 26! − 1.7. (a) 2791. (c) 112135 25032 442.9. (a) 31. (c) 14.10. (a) n = 11 · 41. (c) n = 8779 · 4327.Chapter 7. Algebraic Coding Theory2. (0000) /∈ C.3. (a) 2. (c) 2.4. (a) 3. (c) 4.6. (a) d min = 2. (c) d min = 1.