CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The square-affine group of F_7 is 2-homogeneous but not 2-transitive

Statement refuted

Every $2$ -homogeneous action is $2$ -transitive.

Facts & Assumptions

Given: The subgroup $G : = {x \mapsto a x + b : a \in {1, 2, 4}, b \in F_{7}}$ of the affine group of $F_{7}$ .

[L1]

A $2$ -homogeneous action sends any $2$ -element subset to any other, while a $2$ -transitive action sends any ordered pair of distinct points to any other ordered such pair (k-transitive and k-homogeneous actions).

Counterexample

technique · direct

1.1L1algebra

The nonzero squares in $F_{7}$ are ${1, 2, 4}$ , and every nonzero element is either a square or the negative of a square. So for any distinct $x, y, u, v \in F_{7}$ , after possibly swapping $u$ and $v$ there is $a \in {1, 2, 4}$ with $a (y - x) = v - u$ . Then the affine map $z \mapsto a (z - x) + u$ lies in $G$ and sends ${x, y}$ to ${u, v}$ . Thus the action is $2$ -homogeneous.

1.2L1algebra

Every element of $G$ multiplies differences by a square: if $g (z) = a z + b$ , then $g (y) - g (x) = a (y - x)$ with $a \in {1, 2, 4}$ . Since $1$ is a square and $3$ is not, no element of $G$ sends the ordered pair $(0, 1)$ to $(0, 3)$ . So the action is not $2$ -transitive.

2.1step 1.1step 1.2∎

Step 1.1 gives $2$ -homogeneity while step 1.2 denies $2$ -transitivity, refuting the statement.

Depends on

k-transitive and k-homogeneous actions

Used by

FALSE: every 2-homogeneous action is 2-transitive False statement

Dependency tree · two levels

2 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

P. J. Cameron, Permutation Groups, Chapter 2 (standard reference, not scraped)
K. Conrad, Transitive Group Actions (standard reference, not scraped)