False statementConstruction: 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.

FALSE: groups with isomorphic Cayley graphs are isomorphic

Statement refuted

groups with isomorphic Cayley graphs are isomorphic.

Facts & Assumptions

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

[F1]

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set ${{g, g s} : g \in G, s \in (S \cup S^{- 1}) ∖ {e}}$ (The Cayley graph of a group with respect to a subset).

[L1]

On a finite vertex set $V$ , the empty graph has edge set $\emptyset$ and the complete graph $K_{V}$ has edge set $[V]^{2}$ . When $V$ is an $n$ -element labelled set, these are also denoted ${\overline{K}}_{n}$ and $K_{n}$ . (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[L2]

If $G = ⟨ g ⟩$ is cyclic, then exactly one of the following applies: (Every cyclic group is isomorphic to $(Z, +)$ or to $(Z / n, +)$ for its finite order $n \geq 1$ ).

[L3]

A graph isomorphism is a bijection $φ : V \to W$ such that, for all distinct $u, v \in V$ , (Graph isomorphisms, automorphisms and graph complements).

Refutation

technique · contradiction

1.1

The claim asserts that the isomorphism type of some Cayley graph determines the group.

F1assume-contra

2.1

Taking the whole group as generating set, both the symmetric group on three letters and the cyclic group of order six give the complete graph on six vertices, and those groups are not isomorphic.

F1L1L2L3step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)