DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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The cycle index of a finite permutation group

Definition

Let a finite group $G$ act on a finite set $X$ of size $n$ . For $g \in G$ and $1 \leq d \leq n$ , let $j_{d} (g)$ be the number of $d$ -cycles in the permutation of $X$ induced by $g$ . Then

$\sum_{d = 1}^{n} d j_{d} (g) = n,$

so the monomial

$\prod_{d = 1}^{n} s_{d}^{j_{d} (g)}$

lies in the polynomial ring $Q [s_{1}, \dots, s_{n}]$ (Polynomial rings in finitely many commuting indeterminates by iteration).

The cycle index of the permutation action $G ↷ X$ is

$Z_{G} (s_{1}, \dots, s_{n}) : = \frac{1}{∣ G ∣} \sum_{g \in G} \prod_{d = 1}^{n} s_{d}^{j_{d} (g)} .$

When the acting set is clear from context, this polynomial is also written $Z (G)$ .

Depends on

Used by

Colourings, weight functions, and the pattern inventory Definition
The cycle-index series of a graded family of Sₙ-actions Definition
Colourings of the faces of a cube up to rotation Example
FALSE: nonisomorphic groups acting on finite sets always have different cycle indices False statement
FALSE: the cycle index of a permutation action determines the abstract group False statement
The cycle index of Sₙ is the sum over cycle types Theorem
The cycle index of the cyclic group Cₙ Theorem
The cycle index of the dihedral group D₂ₙ Theorem

Dependency tree · two levels

9 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)
Eric W. Weisstein, Cycle Index, Wolfram MathWorld (standard reference, not scraped)