False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.
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Differential at the identity determines a homomorphism from a disconnected Lie group

Statement refuted

A homomorphism from a disconnected Lie group is determined by its differential at the identity.

Facts & Assumptions

Given: The finite discrete Lie group $G = Z / 2 Z$ .

[F1]

A smooth group homomorphism is a Lie-group homomorphism. Lie-group homomorphism, isomorphism, and automorphism.

[F2]

A discrete finite group is a zero-dimensional Lie group: every map between discrete charts is smooth. Lie group.

Refutation

technique · counterexample

1.1

Let $F : G \to G$ be the identity and let $H : G \to G$ be the trivial homomorphism. Both are smooth by discreteness and hence are Lie-group homomorphisms by [F1]–[F2], but $F (1) = 1 \neq 0 = H (1)$ .

F1F2

2.1

The tangent space of a zero-dimensional manifold at every point is the zero vector space. Thus $d F_{0}$ and $d H_{0}$ are both the unique map $0 \to 0$ , despite $F \neq H$ .

F2step 1.1

3.1

The witness is nonempty, zero-dimensional, and disconnected; it shows exactly why connectedness cannot be dropped. No metric, degeneracy beyond the deliberately zero tangent space, interval, endpoint, choice, or biconditional occurs.

F1F2step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)