9.1 The Jordan-Hölder Theorem
1.
a. If C8 = 
g 
, o(g) = 8, then o(g2) = 4 and o(g4) = 2. It follows that C8 ⊃ g2 ⊃ g4 ⊃ {1} is a composition series. Hence length C8 = 3 and the factors are C2, C2, C2.
g , o(g) = 8, then o(g2) = 4 and o(g4) = 2. It follows that C8 ⊃ g2 ⊃ g4 ⊃ {1} is a composition series. Hence length C8 = 3 and the factors are C2, C2, C2.
c. We have D4 = {1, a, a2, a3, b, ba, ba2, ba3} where o(a) = 4, o(b) = 2, aba = b. Hence D4 ⊃ a ⊃ a2 ⊃ {1} is a composition series so length D4 = 3 and the factors are C2, C2, C2.
a ⊃ a2 ⊃ {1} is a composition series so length D4 = 3 and the factors are C2, C2, C2.
e. Q = { ± 1, ± i, ± j, ± k} and o(i) = 4. If M = i , K = − 1 , then Q ⊃ M ⊃ K ⊃ {1} is a composition series. ...
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