CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An extraspecial group of order $32$ decomposes both as two quaternion factors and as two dihedral factors

Statement refuted

The central-product decomposition of an extraspecial group into factors of order $p^{3}$ is unique.

Facts & Assumptions

Given: The proposed claim together with the witness named in the Statement refuted.

[L1]

Subgroups $G_{1}, \dots, G_{r}$ of $G$ form an internal central product when they generate $G$ and $[G_{i}, G_{j}] = 1$ for $i \neq j$ (Internal central products of a finite family of subgroups).

[L2]

Subgroups $G_{1}, \dots, G_{r}$ form an internal central product of $G$ if and only if the multiplication map $G_{1} \times \dots \times G_{r} \to G$ is a surjective homomorphism each of whose factors meets its kernel trivially (Internal central products are the images of external ones).

[L3]

$Q_{8} \circ Q_{8} ≅ Dih (C_{4}) \circ Dih (C_{4})$ ( $Q_{8} \circ Q_{8} ≅ Dih (C_{4}) \circ Dih (C_{4})$ ).

[L4]

For each $n \geq 1$ there are exactly two extraspecial groups of order $2^{1 + 2 n}$ up to isomorphism, with $2^{2 n} + 2^{n}$ and $2^{2 n} - 2^{n}$ solutions of $x^{2} = 1$ (For each $n \geq 1$ there are exactly two extraspecial groups of order $2^{1 + 2 n}$ ).

[L5]

$Q_{8} : = {1, - 1, i, - i, j, - j, k, - k} \subseteq H^{\times} .$ (The quaternion group $Q_{8} = {\pm 1, \pm i, \pm j, \pm k}$ inside the nonzero quaternions).

[L6]

$Dih (A) = A ⋊ C_{2} .$ ( The generalized dihedral group $Dih (A) = A ⋊ C_{2}$ for an abelian group $A$ ).

Counterexample

technique · constructive

1.1

Inside one extraspecial group of order thirty-two, exhibit two quaternion subgroups and two dihedral subgroups, using the explicit generators of the cited isomorphism.

L3L5L6construct

2.1

Each pair satisfies the internal central-product conditions: elementwise commuting, intersection the centre, and generating the group.

L1L2step 1.1

3.1

So the isomorphism type of the factors is not determined by the group, although the group itself is one of the two given by the classification.

L3L4step 2.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 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

D. A. Craven, The Theory of p-Groups (Hilary Term 2008), 48 pp. (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, 62 pp. (standard reference, not scraped)
D. Kaur and A. Kulshrestha, Characters of real special 2-groups (arXiv:1510.06583v1) (standard reference, not scraped)

An extraspecial group of order 3232 decomposes both as two quaternion factors and as two dihedral factors