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

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The natural action of $S_3$ on three points is faithful and transitive but not free

Statement refuted

False claim. Every faithful transitive group action is free.

Facts & Assumptions

Given: The symmetric group $S_3=\operatorname{Sym}(\{1,2,3\})$ acting on $X=\{1,2,3\}$ by evaluation.

[L1]

An action is transitive when every point can be carried to every other and faithful when only the identity fixes every point (Left group actions, transitive actions, and faithful actions).

[L2]

An action is free when no nonidentity element fixes any point (A free group action has no nonidentity element fixing a point).

[L3]

Stabilizers record the elements fixing a chosen point (The orbit $G\cdot x$ and stabilizer $G_x$ of a point in a group action).

[L4]

The symmetric group consists of all bijections of the set (The symmetric group $\operatorname{Sym}(X)$ : the bijections of a set $X$ under composition).

[L5]

The symmetric group is a group under composition ( $\operatorname{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements).

Counterexample

technique · direct

1.1

Evaluation satisfies $e(i)=i$ and $(\sigma\tau)(i)=\sigma(\tau(i))$ , so [L4] and [L5] give an action of $S_3$ on $X$ .

L4L5

2.1

For any $i,j\in X$ , a permutation carries $i$ to $j$ , so the action is transitive. If a permutation fixes all three points, it is the identity function, so the action is faithful.

step 1.1L1L4

3.1

The nonidentity transposition $(2\,3)$ fixes $1$ , so $G_1$ is nontrivial and the action is not free by [L2] and [L3]. Thus the false claim fails.

step 2.1L2L3∎

Depends on

Used by

S₃ acting on three points has |X|=3 and |X^S₃|=0, so the fixed-point congruence modulo 2 fails without the p-group hypothesis Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 11 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

T. W. Judson, Abstract Algebra: Theory and Applications, 14.1 (standard reference, not scraped)

The natural action of S3S_3S3​ on three points is faithful and transitive but not free