TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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.

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.

If a finite $p$ -group $P$ acts on a finite set $X$ , then $∣ X ∣ \equiv ∣ X^{P} ∣ (m o d p)$

Statement

If a finite $p$ -group $P$ acts on a finite set $X$ , then

$∣ X ∣ \equiv ∣ X^{P} ∣ (m o d p) .$

Facts & Assumptions

Given: A finite $p$ -group $P$ acting on a finite set $X$ .

[L1]

A finite $p$ -group has prime-power order (A finite $p$ -group has order $p^{n}$ for a prime $p$ and some $n \in N$ ).

[L2]

The global fixed-point set is $X^{P} = {x : g \cdot x = x for every g \in P}$ (The fixed-point sets $X^{g}$ and $X^{G}$ of a group action).

[L3]

The $P$ -orbits partition $X$ (The orbits of a group action are the equivalence classes of $x \sim y$ iff $y = g \cdot x$ for some $g$ , and hence partition the acted-on set).

[L4]

An orbit has size $∣ P \cdot x ∣ = [P : P_{x}]$ (Orbit-stabiliser cardinality: $∣ G \cdot x ∣ = [G : G_{x}]$ whenever either side is finite, and $∣ G ∣ = ∣ G_{x} ∣ ∣ G \cdot x ∣$ for finite $G$ ).

[L5]

Every subgroup of $P$ has prime-power order (Every subgroup of a finite $p$ -group has order a power of $p$ ).

[L6]

The congruence $a \equiv b (m o d p)$ means that $p$ divides $a - b$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

[L7]

A finite partition has total cardinality equal to the sum of its block cardinalities (The sum rule: a finite disjoint union is finite with $∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣$ and $∣ ⋃_{i \in I} A_{i} ∣ = \sum_{i \in I} ∣ A_{i} ∣$ , and a sum over a finite index set splits along a partition).

[L8]

Finite sums over finite index sets are well-defined (The sum $\sum_{i \in S} a_{i}$ over a finite index set, and its product form).

Proof

technique · direct

1.1

By [L3], $X$ is the disjoint union of its $P$ -orbits. An orbit is a singleton exactly when its point is fixed by every element of $P$ , so the singleton orbits are indexed by $X^{P}$ .

L2L3

1.2

For a non-singleton orbit $P \cdot x$ , the stabilizer $P_{x}$ is proper. By [L1], [L4], and [L5], its index is a positive power of $p$ , so $p$ divides $∣ P \cdot x ∣$ .

L1L4L5

2.1

Applying [L7] and [L8] to the orbit partition, every non-singleton orbit contributes a multiple of $p$ and the singleton orbits contribute $∣ X^{P} ∣$ . Thus $p$ divides $∣ X ∣ - ∣ X^{P} ∣$ , which is the asserted congruence by [L6].

step 1.1step 1.2L6L7L8∎

Depends on

Used by

Dependency tree · two levels

37 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

K. Conrad, Group Actions, Theorem 4.1 (standard reference, not scraped)

If a finite pp-group PP acts on a finite set XX, then ∣X∣≡∣XP∣(modp)∣X∣≡∣XP∣(modp)