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The maximal subgroups of the dihedral group of order eight as Frattini hyperplanes

Example

For $D = Dih (C_{4}) = ⟨ r, s ⟩$ , the maximal subgroups are

$⟨ r ⟩, ⟨ r^{2}, s ⟩, ⟨ r^{2}, r s ⟩ .$

Modulo $Φ (D) = ⟨ r^{2} ⟩$ , these are the hyperplanes of $(Z / 2)^{2}$ .

Facts & Assumptions

Given: The dihedral group $D = Dih (C_{4})$ of order eight.

[L1]

For $D = Dih (C_{4})$ one has $Φ (D) = ⟨ r^{2} ⟩$ , and for the dihedral group of order eight and $Q_{8}$ the Frattini subgroup has order two and the Frattini quotient is $(Z / 2)^{2}$ (The Frattini subgroups of the dihedral and quaternion groups of order eight).

[L2]

If $π : P \to E = P / Φ (P)$ is the quotient map, the maximal subgroups of $P$ are exactly $π^{- 1} (\ker λ) = \ker (λ \circ π)$ for nonzero $F_{p}$ -linear homomorphisms $λ : E \to Z / p$ (Maximal subgroups of a finite $p$ -group are the inverse images of Frattini hyperplanes).

Verification

technique · direct

1.1

The three displayed subgroups have order four, are distinct, and contain $Φ (D) = ⟨ r^{2} ⟩$ from [L1]. Since $D$ has order eight, each is maximal.

givenL1algebra

2.1

Use the quotient basis $(r Φ (D), s Φ (D))$ . The quotient images of the displayed subgroups are the lines spanned by $(1, 0)$ , $(0, 1)$ , and $(1, 1)$ , which are respectively the kernels of $(a, b) \mapsto b$ , $(a, b) \mapsto a$ , and $(a, b) \mapsto a + b$ . These are precisely the hyperplanes described by [L2].

step 1.1L2algebra∎

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Sources

D. A. Craven, The Theory of p-Groups, §2.2 (standard reference, not scraped)