LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Positive-degree homology of a finite group is order-torsion

Statement

For finite $G$ and $n > 0$ , $∣ G ∣$ annihilates $H_{n} (G; Z)$ .

Facts & Assumptions

Given: Let $G$ be finite and $n > 0$ .

[L1]

Weibel's Theorem 6.5.8 states that $∣ G ∣$ annihilates $H_{n} (G; A)$ for every finite group $G$ , every $G$ -module $A$ , and every $n > 0$ .

Proof

technique · direct

1.1

The cited theorem of Weibel states that, for a finite group $G$ , multiplication by $∣ G ∣$ annihilates $H_{n} (G; A)$ for every $n > 0$ and every $G$ -module $A$ . Apply it to the trivial module $A = Z$ .

L1given

2.1

Thus multiplication by $∣ G ∣$ is zero on $H_{n} (G; Z)$ , as claimed.

step 1.1∎

Depends on

Group homology as a derived functor

Used by

Multiplier of a finite group is finite Lemma

Dependency tree · two levels

5 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 6, Theorem 6.5.8 (standard reference, not scraped)