ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The dihedral group of order eight is a split extension of C_4 by C_2

Example

Let $D_{4} = ⟨ r, s ∣ r^{4} = s^{2} = 1, s r s^{- 1} = r^{- 1} ⟩$ . Then

$1 \to ⟨ r ⟩ \to D_{4} \to ⟨ s ⟩ \to 1$

is a split extension of $C_{4}$ by $C_{2}$ .

Facts & Assumptions

Given: The library convention in which $D_{n}$ is the dihedral group of order $2 n$ , and the standard presentation of $D_{4}$ .

[L1]

The dihedral group of order eight is the semidirect product $C_{4} ⋊ C_{2}$ with inversion action ( $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations).

[L2]

A complement to the kernel is equivalent to a split extension (A group extension splits exactly when it has a complement or a compatible semidirect-product model, and a kernel retraction forces a direct product).

Verification

technique · direct

1.1

By [L1], the rotation subgroup $⟨ r ⟩$ is a normal copy of $C_{4}$ and the reflection subgroup $⟨ s ⟩$ is a copy of $C_{2}$ . Their intersection is trivial and they generate $D_{4}$ .

givenL1

2.1

Thus $⟨ s ⟩$ is a complement to $⟨ r ⟩$ in $D_{4}$ , so [L2] gives the displayed split extension of $C_{4}$ by $C_{2}$ .

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Group Theory (standard reference, not scraped)