CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The same middle group can support inequivalent extension maps

Statement refuted

Two extensions are equivalent whenever their middle groups are isomorphic.

Facts & Assumptions

Given: The middle group $E = C_{9} = Z / 9 Z$ , the quotient map

$π (\overline{x}) = \overline{x} (m o d 3),$

and the two kernel embeddings

$i_{1} (\overline{a}) = 3 \overline{a}, i_{2} (\overline{a}) = 6 \overline{a} .$

[L1]

The false statement above is the claim to be refuted (FALSE: equivalent extensions mean only that the middle groups are isomorphic).

Counterexample

technique · direct

1.1

The map $π$ is surjective, with kernel ${0, \overline{3}, \overline{6}}$ . Both $i_{1}$ and $i_{2}$ identify $C_{3}$ with that kernel, so $1 \to C_{3} \overset{i_{1}}{\to} E \overset{π}{\to} C_{3} \to 1, 1 \to C_{3} \overset{i_{2}}{\to} E \overset{π}{\to} C_{3} \to 1$ are two extensions of $C_{3}$ by $C_{3}$ with the same middle group $E$ .

givenalgebra

2.1

Any automorphism of $E = C_{9}$ has the form $ϕ_{u} (\overline{x}) = u \overline{x}$ with $u \in (Z / 9 Z)^{\times}$ . If an extension equivalence existed, it would satisfy $ϕ \circ i_{1} = i_{2}$ and $π \circ ϕ = π$ . The first identity gives $3 u \equiv 6 (m o d 9)$ , so $u \equiv 2 (m o d 3)$ . The second gives $u \equiv 1 (m o d 3)$ . This contradiction shows that no extension equivalence can exist.

step 1.1algebra

3.1

Thus the same middle group supports two inequivalent extension structures, refuting [L1].

L1step 2.1∎

Depends on

FALSE: equivalent extensions mean only that the middle groups are isomorphic

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)