False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-26

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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.

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FALSE: for each $n \geq 1$ there is exactly one extraspecial group of order $p^{1 + 2 n}$ up to isomorphism

Statement refuted

for each $n \geq 1$ there is exactly one extraspecial group of order $p^{1 + 2 n}$ up to isomorphism.

Facts & Assumptions

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

[L1]

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 $2^{1 + 2 n}$ ).

[L2]

For odd $p$ and each $n \geq 1$ there are exactly two extraspecial groups of order $p^{1 + 2 n}$ up to isomorphism, one of exponent $p$ and one of exponent $p^{2}$ (For odd $p$ and each $n \geq 1$ there are exactly two extraspecial groups of order $p^{1 + 2 n}$ , distinguished by their exponent).

[L3]

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G \approx n$ for some $n \in N$ . (The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity).

[L4]

For a finite group $G$ , its exponent is $\exp (G) = \min {n \in N : n > 0 and g^{n} = e for every g \in G} .$ The set is nonempty by, and gives its least member; powers use. (The exponent of a finite group).

Refutation

technique · contradiction

1.1assume-contra

The claim asserts uniqueness up to isomorphism at each order $p^{1 + 2 n}$ .

2.1L1L2L3L4step 1.1discharge-contradiction∎

Both classifications produce two types at each such order, separated by the count of square roots of the identity when $p = 2$ and by the exponent when $p$ is odd.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

37 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)

FALSE: for each n≥1n≥1 there is exactly one extraspecial group of order p1+2np1+2n up to isomorphism