2.9 Factor Groups
1.
a. D6 = {1, a, a2, a3, a4, a5, b, ba, ba2, ba3, ba4, ba5}; K = Z(D6) = {1, a3} by Exercise 26 §2.6. The cosets are
We have 
, 
and KaKbKa = Kaba = Kb. Hence D6/K ≅ D3.
, and KaKbKa = Kaba = Kb. Hence D6/K ≅ D3.
c. G = A × B, K = {(a, 1) 
a 
A}. Note K G because (x, y)−1(a, 1)(x, y) = (x−1ax, 1) K. If b B, a A, note that (ab)(1, b)−1 K, so
a A}. Note K G because (x, y)−1(a, 1)(x, y) = (x−1ax, 1) K. If b B, a A, note that (ab)(1, b)−1 K, so
Thus G/K = {K(1, b) b B}. Observe that K(1, b)K(1, b′) = K(1, bb′) gives ...
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