ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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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.
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An involution on five points has three fixed points and one two-point orbit, verifying $5 \equiv 3 (m o d 2)$

Example

Let $Z / 2$ act on $X = {1, 2, 3, 4, 5}$ so that its nonidentity element interchanges $1$ and $2$ and fixes $3, 4, 5$ . Then $∣ X^{Z / 2} ∣ = 3$ and $5 \equiv 3 (m o d 2)$ .

Facts & Assumptions

Given: The additive group $P = Z / 2$ and the displayed permutation of $X$ .

[L1]

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

[L2]

The residue classes modulo $2$ form a group (For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

[L3]

The two classes are represented by $0$ and $1$ (For $n \geq 1$ , every class in $Z / n$ has one representative $r$ with $0 \leq r < n$ , so $∣ Z / n ∣ = n$ ; while $Z / 0$ is in bijection with $Z$ ).

[L4]

Congruence modulo $2$ means divisibility of the difference by $2$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

Verification

technique · direct

1.1

Map $0$ to the identity permutation and $1$ to $(1 2)$ . Since $(1 2)^{2} = e$ , [L2] and [L3] give an action of $P$ on $X$ .

L2L3

2.1

Its orbit partition is ${1, 2}, {3}, {4}, {5}$ , and the global fixed set is ${3, 4, 5}$ .

step 1.1L1

3.1

Hence $∣ X ∣ = 5$ , $∣ X^{P} ∣ = 3$ , and $5 - 3 = 2$ is divisible by $2$ , verifying [L1] by [L4].

step 2.1L1L4algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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)

An involution on five points has three fixed points and one two-point orbit, verifying 5≡3(mod2)5≡3(mod2)