False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)audited 2026-08-28

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FALSE: the cycle index of a permutation action determines the abstract group

Statement

False claim: once one knows the cycle index of a finite permutation action, the acting group is determined up to isomorphism.

Facts & Assumptions

Given: the four-point set $X = {1, 2, 3, 4}$ and the permutation $τ = (1 2) (3 4)$ .

[F1]

The cycle index averages the cycle monomials of the acting permutations (The cycle index of a finite permutation group).

Refutation

technique · direct

1.1

Let $C_{4} = ⟨ g ⟩$ act on $X$ through the quotient map $g \mapsto τ$ . Then two elements act as the identity and two act as $τ$ , so [F1] gives the cycle index $\frac{1}{4} (2 s_{1}^{4} + 2 s_{2}^{2}) = \frac{1}{2} (s_{1}^{4} + s_{2}^{2})$ .

F1algebra

2.1

Let $V_{4}$ act on $X$ through a quotient onto ${1, τ}$ with kernel of size $2$ . Again two elements act as the identity and two act as $τ$ , so the same calculation gives the same cycle index $\frac{1}{2} (s_{1}^{4} + s_{2}^{2})$ .

F1step 1.1algebra

3.1

The abstract groups $C_{4}$ and $V_{4}$ are not isomorphic, so equal cycle index does not determine the acting group. Therefore the displayed claim is false.

step 1.1step 2.1∎

Depends on

Used by

C₄ and V₄ can share the same cycle index through nonfaithful actions Counterexample

Dependency tree · two levels

6 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

Ben Lynn, Polya Theory: The Cycle Index Polynomial (standard reference, not scraped)