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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A finite $p$ -group action on $X$ has a global fixed point whenever $p ∤ ∣ X ∣$

Statement

Let a finite $p$ -group $P$ act on a finite set $X$ . If $p ∤ ∣ X ∣$ , then $X^{P} \neq \emptyset$ ; equivalently, the action has a point fixed by every element of $P$ .

Facts & Assumptions

Given: A finite $p$ -group $P$ acting on a finite set $X$ , with $p ∤ ∣ X ∣$ .

[L1]

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

[L2]

The set $X^{P}$ consists of the points fixed by every element of $P$ (The fixed-point sets $X^{g}$ and $X^{G}$ of a group action).

[L3]

Congruence modulo $p$ means divisibility of the difference by $p$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

[L4]

A prime is positive and greater than $1$ (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ ).

Proof

technique · contradiction

1.1

Suppose, for contradiction, that $X^{P} = \emptyset$ . Then $∣ X^{P} ∣ = 0$ .

assume-contraL2

2.1

By [L1] and [L3], $p$ divides $∣ X ∣ - ∣ X^{P} ∣ = ∣ X ∣$ , contradicting the hypothesis.

step 1.1L1L3

3.1

Therefore $X^{P}$ is nonempty, and any of its elements is a global fixed point by [L2].

step 2.1L2L4discharge-contradiction∎

Depends on

Used by

Hall–Burnside: coprime automorphisms are detected on the Frattini quotient Theorem

Dependency tree · two levels

25 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, Corollary 4.2 (standard reference, not scraped)

A finite pp-group action on XX has a global fixed point whenever p∤∣X∣p∤∣X∣