DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

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Paradoxical decompositions of groups

Definition

Let $G$ be a group. A paradoxical decomposition of $G$ consists of pairwise disjoint subsets

$A_{1}, \dots, A_{m}, B_{1}, \dots, B_{n} \subseteq G$

and group elements $a_{1}, \dots, a_{m}, b_{1}, \dots, b_{n} \in G$ such that

$G = ⨆_{i = 1}^{m} A_{i} ⊔ ⨆_{j = 1}^{n} B_{j} = ⨆_{i = 1}^{m} a_{i} A_{i} = ⨆_{j = 1}^{n} b_{j} B_{j} .$

Thus the group is partitioned into finitely many pieces, and each of the two subfamilies can be translated to cover the whole group again.

Depends on

Group and abelian group

Used by

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

C. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)
Cornelia Drutu and Michael Kapovich, Lectures on Geometric Group Theory (standard reference, not scraped)