ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The three maximal abelian subgroups of $Dih (C_{4})$ have order four, as the general bound predicts

Example

The three maximal abelian subgroups of $Dih (C_{4})$ have order four, as the general bound predicts.

Facts & Assumptions

Given: The objects and hypotheses in the Example.

[L1]

Every maximal abelian subgroup of an extraspecial $p$ -group of order $p^{1 + 2 n}$ has order $p^{1 + n}$ (In an extraspecial $p$ -group of order $p^{1 + 2 n}$ every maximal abelian subgroup has order $p^{1 + n}$ ).

[L2]

The generalized dihedral group $Dih (C_{4})$ and the quaternion group $Q_{8}$ are extraspecial of order $8$ , with six and two solutions of $x^{2} = 1$ respectively ( $Dih (C_{4})$ and $Q_{8}$ are extraspecial of order $8$ , with six and two solutions of $x^{2} = 1$ respectively).

[L3]

For $n \geq 1$ , $Dih (C_{n}) = C_{n} ⋊ C_{2}$ has order $2 n$ , and with $C_{n} = ⟨ r ⟩$ and $C_{2} = ⟨ s ⟩$ one has $r^{n} = s^{2} = 1$ , $s r s^{- 1} = r^{- 1}$ , and every element has a unique form $r^{i}$ or $r^{i} s$ with $0 \leq i < n$ ( $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations).

[L4]

$∣ G ∣ = [G : H] ∣ H ∣ .$ (Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

Verification

technique · direct

1.1

Write $D = Dih (C_{4}) = ⟨ r, s ⟩$ . Its three subgroups of order four are $⟨ r ⟩$ , $⟨ r^{2}, s ⟩$ , and $⟨ r^{2}, r s ⟩$ .

L2L3

2.1

The subgroup $⟨ r ⟩$ is cyclic, while $⟨ r^{2}, s ⟩$ and $⟨ r^{2}, r s ⟩$ are Klein four groups because $r^{2}$ is central of order two and both $s$ and $r s$ are involutions. Thus all three are abelian. Each has order four in the order-eight group $D$ ; any larger subgroup would be the whole group, which is nonabelian by [L2], so each is maximal among abelian subgroups.

L2L3L4step 1.1

3.1

Their common order four equals $p^{1 + n}$ at $p = 2$ and $n = 1$ , exactly as [L1] predicts. Their pairwise intersections are all $⟨ r^{2} ⟩ = Z (D)$ ; and $⟨ r ⟩ ⟨ r^{2}, s ⟩ = D$ , $⟨ r ⟩ ⟨ r^{2}, r s ⟩ = D$ , and $⟨ r^{2}, s ⟩ ⟨ r^{2}, r s ⟩ = D$ because in each case the two displayed subgroups contain generators $r$ and $s$ of $D$ .

L1L2L3step 2.1∎

Depends on

Used by

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Sources

D. A. Craven, The Theory of p-Groups (Hilary Term 2008), 48 pp. (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, 62 pp. (standard reference, not scraped)
D. Kaur and A. Kulshrestha, Characters of real special 2-groups (arXiv:1510.06583v1) (standard reference, not scraped)

The three maximal abelian subgroups of Dih⁡(C4)Dih(C4​) have order four, as the general bound predicts