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

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FALSE: every split group extension is a direct product

Statement

Every split group extension is an internal direct product.

Facts & Assumptions

Given: The extension $1 \to C_{4} \to D_{4} \to C_{2} \to 1$ coming from the rotation subgroup and a reflection complement, in the convention that $D_{n}$ has order $2 n$ .

[L1]

A split extension is direct exactly when the complement centralizes the kernel (A split extension is a direct product exactly when its complement centralizes the kernel).

[L2]

A group extension splits if and only if its kernel has a complement (A group extension splits exactly when it has a complement or a compatible semidirect-product model, and a kernel retraction forces a direct product).

Refutation

technique · direct

1.1givenL2

In $D_{4} = ⟨ r, s ∣ r^{4} = s^{2} = 1, s r s^{- 1} = r^{- 1} ⟩$ , the subgroup $⟨ r ⟩ ≅ C_{4}$ is normal and $⟨ s ⟩ ≅ C_{2}$ is a complement, so the reverse implication of [L2] makes the extension split.

2.1L1step 1.1∎

But $s r s^{- 1} = r^{- 1} \neq r$ , so the complement does not centralize the kernel. By [L1], the decomposition is not a direct product. Hence the claim is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)