TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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Positive integral homology is annihilated by the group order

Statement

For a finite group G and $n > 0$ , $∣ G ∣ H_{n} (G; Z) = 0$ , with trivial integral coefficients. Derived homology uses the inherited DC and supplied-resolution convention.

Facts & Assumptions

Given: A finite group G and integer n>0.

[F1]

Inclusion after transfer multiplies by the finite index (Corestriction after transfer multiplies by the index).

[F2]

Normalized diagonal bars compute group homology (Diagonal bar coinvariants compute group homology).

Proof

1.1

For the trivial group, every tuple of length at least two has adjacent equal vertices. Its normalized chain groups in positive degrees are consequently zero, so $H_{n} (1; Z) = 0$ for n>0.

F2given

2.1

Apply transfer to $1 \leq G$ . Its index is $∣ G ∣$ and the composite factors through the zero group of step 1.1; F1 identifies this composite with multiplication by $∣ G ∣$ . Hence it is zero. If G=1, this says the positive homology itself is zero. Degree zero is excluded: there it is $Z$ .

F1step 1.1algebra∎

Depends on

Used by

Integral homology of a finite cyclic group Example

Dependency tree · two levels

9 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

Loh, Group Cohomology, Definitions 1.7.12–13 and Theorem 1.7.15 pp.63–64; Theorem 3.2.18 pp.129–132 (standard reference, not scraped)
Weibel, An Introduction to Homological Algebra, Chapter 6, Sections 6.4–6.8 (standard reference, not scraped)