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Abstract Algebra

Exercise 2.5.16 (Every element of order $2$ in $QD_{16}$ is in $\langle \sigma^2, \tau \rangle$)

Use the lattice of subgroups of the quasidihedral group of order 16 to show that every element of order 2 is contained in the proper subgroup ⟨ τ , σ 2 ⟩ (cf. Exercise 11).

Answers

Proof. We obtained this lattice in Exercise 11:

Every element of order 2 generates a subgroup of order 2, which is by this lattice in the list

⟨ σ 4 ⟩ , ⟨ τ ⟩ , ⟨ τ σ 4 ⟩ , ⟨ τ σ 6 ⟩ , ⟨ τ σ 2 ⟩ .

All are subgroups of ⟨ σ 2 , τ ⟩ . So every element of order 2 is contained in the proper subgroup ⟨ τ , σ 2 ⟩ □

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2025-11-11 17:48
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