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

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The cyclic group Z/9 supports inequivalent extensions of C_3 by C_3

Statement refuted

If two extensions of the same kernel and quotient have isomorphic middle groups, then they are equivalent as extensions.

The cyclic group $Z / 9 Z$ supports two inequivalent extensions of $C_{3}$ by $C_{3}$ .

Facts & Assumptions

Given: The additive cyclic group $E = Z / 9 Z$ , together with $N = Q = Z / 3 Z$ and the inclusion $j : N \to E$ given by $j (\overset{ˉ}{a}) = \overline{3 a}$ .

[L1]

The false claim under refutation is FALSE: isomorphic middle groups force equivalent extensions with fixed kernel and quotient.

Counterexample

technique · direct

1.1

Let $π_{1}, π_{2} : E \to Q$ be the surjective homomorphisms $π_{1} (\overset{ˉ}{x}) = \overset{ˉ}{x} (m o d 3), π_{2} (\overset{ˉ}{x}) = 2 \overset{ˉ}{x} (m o d 3) .$ Both have kernel $im j = {\overset{ˉ}{0}, \overset{ˉ}{3}, \overset{ˉ}{6}}$ , so $0 \to N \overset{j}{\to} E \overset{π_{1}}{\to} Q \to 0, 0 \to N \overset{j}{\to} E \overset{π_{2}}{\to} Q \to 0$ are two extensions of $Q$ by $N$ with the same middle group $E$ .

givenalgebra

2.1

The middle groups are literally identical, but the two extension structures are not equivalent. Indeed, any automorphism of $E = Z / 9 Z$ is multiplication by a unit $u \in {1, 2, 4, 5, 7, 8}$ . If it fixed the kernel inclusion, then $u \equiv 1 (m o d 3)$ ; if it also carried $π_{1}$ to $π_{2}$ , then $2 u \equiv 1 (m o d 3)$ , so $u \equiv 2 (m o d 3)$ , impossible. Thus [L1] is false.

L1step 1.1algebra∎

Depends on

FALSE: isomorphic middle groups force equivalent extensions with fixed kernel and quotient

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)