Abstract Algebra Theory and Applications - Computer Science ...

HINTS AND SOLUTIONS 3997. Use the Fundamental Theorem of Finitely Generated Abelian Groups.12. If N and G/N are solvable, then they have solvable seriesThe seriesN = N n ⊃ N n−1 ⊃ · · · ⊃ N 1 ⊃ N 0 = {e}G/N = G n /N ⊃ G n−1 /N ⊃ · · · G 1 /N ⊃ G 0 /N = {N}.G = G n ⊃ G n−1 ⊃ · · · ⊃ G 0 = N = N n ⊃ N n−1 ⊃ · · · ⊃ N 1 ⊃ N 0 = {e}is a subnormal series. The factors of this series are abelian since G i+1 /G i∼ =(G i+1 /N)/(G i /N).16. Use the fact that D n has a cyclic subgroup of index 2.21. G/G ′ is abelian.Chapter 12. Group Actions1. Example 1. 0, R 2 \ {0}.Example 2. X = {1, 2, 3, 4}.2. (a) X (1) = {1, 2, 3}, X (12) = {3}, X (13) = {2}, X (23) = {1}, X (123) =X (132) = ∅. G 1 = {(1), (23)}, G 2 = {(1), (13)}, G 3 = {(1), (12)}.3. (a) O 1 = O 2 = O 3 = {1, 2, 3}.6. (a) O (1) = {(1)}, O (12) = {(12), (13), (14), (23), (24), (34)},O (12)(34) = {(12)(34), (13)(24), (14)(23)},O (123) = {(123), (132), (124), (142), (134), (143), (234), (243)},O (1234) = {(1234), (1243), (1324), (1342), (1423), (1432)}.The class equation is 1 + 3 + 6 + 6 + 8 = 24.8. (3 4 + 3 1 + 3 2 + 3 1 + 3 2 + 3 2 + 3 3 + 3 3 )/8 = 21.11. (1 · 3 4 + 6 · 3 3 + 11 · 3 2 + 6 · 3 1 )/24 = 15.15. (1 · 2 6 + 3 · 2 4 + 4 · 2 3 + 2 · 2 2 + 2 · 2 1 )/12 = 13.17. (1 · 2 8 + 3 · 2 6 + 2 · 2 4 )/6 = 80.22. x ∈ gC(a)g −1 ⇔ g −1 xg ∈ C(a) ⇔ ag −1 xg = g −1 xga ⇔ gag −1 x = xgag −1 ⇔x ∈ C(gag −1 ).Chapter 13. The Sylow Theorems1. If |G| = 18 = 2 · 3 2 , then the order of a Sylow 2-subgroup is 2, and the orderof a Sylow 3-subgroup is 9.If |G| = 54 = 2 · 3 3 , then the order of a Sylow 2-subgroup is 2, and the orderof a Sylow 3-subgroup is 27.