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

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The split extension C_2 × C_2 of C_2 by C_2 is direct

Example

The Klein four group $C_{2} \times C_{2}$ gives a split extension of $C_{2}$ by $C_{2}$ that is already a direct product.

Facts & Assumptions

Given: The external direct product $V = C_{2} \times C_{2}$ .

[L1]

The direct-product criterion says 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]

The external direct product has coordinatewise multiplication (The external direct product $G \times H$ with componentwise multiplication, $G \times H$ is a group with identity $(e_{G}, e_{H})$ , coordinatewise inverses, and homomorphic coordinate projections).

Verification

technique · direct

1.1

Let $N = C_{2} \times {1}$ and $H = {1} \times C_{2}$ . By [L2], these are subgroups of $V$ with trivial intersection and product $V$ , so they define a split extension of $C_{2}$ by $C_{2}$ .

givenL2

2.1

Again by [L2], elements of $N$ and $H$ commute coordinatewise. Therefore the complement $H$ centralizes the kernel $N$ , and [L1] makes the extension direct.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)