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Abstract Algebra

Exercise 3.4.2 (Compositions series for $Q_8$ and $D_8$)

Exhibit all 3 composition series for Q 8 and all 7 composition series for D 8 . List the composition factors in each case.

Answers

Proof. ∙ Composition series for Q 8 :

The lattice of subgroups of Q 8 gives three ways to go from ⟨ 1 ⟩ to Q 8 . When | H i + 1 : H i | = 2 , then H i ⊴ H i + 1 , so we obtain three composition series for Q 8

⟨ 1 ⟩ ⊴ ⟨ − 1 ⟩ ⊴ ⟨ i ⟩ ⊴ Q 8 , ⟨ 1 ⟩ ⊴ ⟨ − 1 ⟩ ⊴ ⟨ j ⟩ ⊴ Q 8 , ⟨ 1 ⟩ ⊴ ⟨ − 1 ⟩ ⊴ ⟨ k ⟩ ⊴ Q 8 . Every composition factor is isomorphic to Z 2 .

∙ Composition series for D 8 :

Similarly,

⟨ 1 ⟩ ⊴ ⟨ 𝑠𝑟 ⟩ ⊴ ⟨ r 2 , 𝑠𝑟 ⟩ ⊴ D 8 , ⟨ 1 ⟩ ⊴ ⟨ s r 3 ⟩ ⊴ ⟨ r 2 , 𝑠𝑟 ⟩ ⊴ D 8 , ⟨ 1 ⟩ ⊴ ⟨ r 2 ⟩ ⊴ ⟨ r 2 , 𝑠𝑟 ⟩ ⊴ D 8 , ⟨ 1 ⟩ ⊴ ⟨ r 2 ⟩ ⊴ ⟨ r ⟩ ⊴ D 8 , ⟨ 1 ⟩ ⊴ ⟨ r 2 ⟩ ⊴ ⟨ r 2 , s ⟩ ⊴ D 8 , ⟨ 1 ⟩ ⊴ ⟨ s r 2 ⟩ ⊴ ⟨ r 2 , s ⟩ ⊴ D 8 , ⟨ 1 ⟩ ⊴ ⟨ s ⟩ ⊴ ⟨ r 2 , s ⟩ ⊴ D 8 ,

and every composition factor is isomorphic to Z 2 . □

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2025-12-07 11:51
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