False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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FALSE: all minimal generating sets of a finite group have the same size

Statement

False claim. All inclusion-minimal generating sets of an arbitrary finite group have the same size.

Facts & Assumptions

Given: The additive group $Z / 6$ .

[F1]

A subset $X$ is a minimal generating set when it generates and no proper subset generates (Minimal generating sets of a group, The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[L1]

The quotient group $(Z, +) / 6 Z$ is the additive group $Z / 6$ (For every $n \in N$ , the congruence-class group $(Z / n, +)$ is the quotient group $(Z, +) / n Z$ ).

Refutation

technique · direct

1.1

The singleton ${[1]}$ generates $Z / 6$ , and its only proper subset is empty, which generates only $[0]$ . Thus it is minimally generating by [F1].

givenF1L1algebra

1.2

The set ${[2], [3]}$ generates because $[2] + [2] + [3] = [1]$ . The singleton ${[2]}$ generates ${[0], [2], [4]}$ and ${[3]}$ generates ${[0], [3]}$ , so neither proper singleton generates; hence the two-element set is also minimal by [F1].

givenF1L1algebra

2.1

Steps 1.1 and 1.2 exhibit minimal generating sets of sizes one and two in the same finite group, refuting the claim.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

K. Conrad, Generating Sets, §6 (standard reference, not scraped)