ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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An involution on five points has three fixed points and one two-point orbit, verifying $5\equiv3\pmod2$

Example

Let $\mathbb Z/2$ act on $X=\{1,2,3,4,5\}$ so that its nonidentity element interchanges $1$ and $2$ and fixes $3,4,5$ . Then $|X^{\mathbb Z/2}|=3$ and $5\equiv3\pmod2$ .

Facts & Assumptions

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

[L1]

A finite $p$ -group action satisfies $|X|\equiv|X^P|\pmod p$ (If a finite $p$ -group $P$ acts on a finite set $X$ , then $|X|\equiv|X^P|\pmod p$ ).

[L2]

The residue classes modulo $2$ form a group (For every natural $n$ , $(\mathbb{Z}/n,+)$ 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 $n\ge 1$ , every class in $\mathbb{Z}/n$ has one representative $r$ with $0\le r<n$ , so $\lvert\mathbb{Z}/n\rvert=n$ ; while $\mathbb{Z}/0$ is in bijection with $\mathbb{Z}$ ).

[L4]

Congruence modulo $2$ means divisibility of the difference by $2$ (Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 79 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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\equiv3\pmod25≡3(mod2)