PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Blocks in a finite transitive action have a common size

Statement

Let $G$ act transitively on a finite set $Ω$ , and let $B$ be a block system. Then any two blocks in $B$ have the same finite cardinality. In particular, if $B \in B$ , then $Ω$ is a disjoint union of finitely many copies of $B$ , so $∣ B ∣$ divides $∣ Ω ∣$ .

Facts & Assumptions

Given: A transitive action of $G$ on a finite set $Ω$ and a block system $B$ .

[L1]

A transitive action sends any chosen point to any other point by some element of $G$ (Left group actions, transitive actions, and faithful actions).

[L2]

A block system is a partition of $Ω$ into blocks, and if $B$ is a block then every translate $g \cdot B$ is again a block (Blocks and block systems for a group action).

Proof

technique · direct

1.1L1choose

Let $B, C \in B$ . Choose $b \in B$ and $c \in C$ . By transitivity there is $g \in G$ with $g \cdot b = c$ .

2.1step 1.1L2

Since $g \cdot b \in (g \cdot B) \cap C$ , the two blocks $g \cdot B$ and $C$ meet. Because $B$ is a partition into blocks, they are equal.

3.1step 2.1algebra

The map $B \to C$ , $x \mapsto g \cdot x$ , is a bijection because every group element acts bijectively on $Ω$ . Hence $∣ B ∣ = ∣ C ∣$ .

4.1step 3.1given∎

As $B$ and $C$ were arbitrary, all blocks in $B$ have the same size. Since the blocks are pairwise disjoint and cover the finite set $Ω$ , the cardinality of any block divides $∣ Ω ∣$ .

Depends on

Used by

A transitive action of prime degree is primitive Corollary

Dependency tree · two levels

3 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

J. S. Milne, Group Theory, Chapter 4 (standard reference, not scraped)
K. Conrad, Transitive Group Actions (standard reference, not scraped)