Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-04
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FALSE: first cohomology classifies all subgroups of a semidirect product

Statement

First cohomology classifies all subgroups of a semidirect product.

Facts & Assumptions

Given: The semidirect product A⋊G.

[L1]

First cohomology classifies complements to the kernel up to kernel conjugacy (First cohomology classifies complements up to kernel conjugacy).

Refutation

technique · direct
1.1

Fact [L1] speaks only about complements to the canonical copy of A, meaning subgroups whose projection to G is an isomorphism.

givenL1
2.1

A subgroup such as the kernel copy A itself or the trivial subgroup does not project isomorphically onto G unless G=1. Therefore such subgroups are outside the classification of [L1]. The claim that all subgroups are classified is false.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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