False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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False: the kernel and quotient determine a group extension up to isomorphism

Statement

False claim: the isomorphism types of the kernel and quotient determine the middle group of a group extension up to isomorphism.

Facts & Assumptions

Given: An odd prime $p$ .

[A1]

The false claim says that any two extensions of $C_{2}$ by $C_{p}$ have isomorphic middle groups.

[L1]

An extension of $H$ by $N$ is a short exact sequence $1 \to N \to G \to H \to 1$ (Group extensions, sections, complements, and split extensions).

[L2]

For $n \geq 1$ the dihedral group $D_{n}$ is $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action; taking $n = p$ gives $D_{p}$ ( $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations).

[L3]

Finite cyclic groups are classified by their order (Every cyclic group is isomorphic to $(Z, +)$ or to $(Z / n, +)$ for its finite order $n \geq 1$ ).

Refutation

technique · direct

1.1L1L3algebra

The cyclic group $C_{2 p}$ contains its index-two subgroup $C_{p}$ , and quotienting by it gives $C_{2}$ . Thus it is the middle group of an extension of $C_{2}$ by $C_{p}$ in the sense of [L1].

1.2L1L2

By [L2], the rotation subgroup $C_{p}$ is normal in $D_{p}$ and the complementary reflection subgroup maps isomorphically to the quotient $C_{2}$ . Hence $D_{p}$ is another middle group for the same kernel and quotient.

2.1step 1.1step 1.2A1L2∎

The group $C_{2 p}$ is abelian, while $D_{p}$ is not: inversion on $C_{p}$ is nontrivial because $p$ is odd. Therefore they are not isomorphic, refuting [A1].

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

Keith Conrad, Semidirect Products (standard reference, not scraped)