TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)audited 2026-08-28

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The cycle index of the cyclic group C_n

Statement

Let $C_{n}$ act on the vertices of a labelled $n$ -gon by rotation, with $n \geq 1$ . Then

$Z (C_{n}) = \frac{1}{n} \sum_{d ∣ n} φ (d) s_{d}^{n / d} .$

Facts & Assumptions

Given: an integer $n \geq 1$ and the rotation action of $C_{n}$ on the vertices of a labelled $n$ -gon.

[A1]

A rotation by $r$ steps sends each vertex $i$ to $i + r (m o d n)$ .

[L1]

Euler's totient satisfies $φ (d) = ∣ (Z / d)^{\times} ∣$ (The unit group $(Z / n)^{\times}$ and Euler's totient $φ (n) = ∣ (Z / n)^{\times} ∣$ for $n \geq 1$ ).

Proof

technique · direct

1.1

A rotation by $r$ steps decomposes the $n$ vertices into $\gcd (n, r)$ cycles, each of length $n / \gcd (n, r)$ . Therefore its cycle-index monomial is $s_{n / \gcd (n, r)}^{\gcd (n, r)}$ .

A1algebra

2.1

Fix a divisor $d$ of $n$ . A rotation contributes the monomial $s_{d}^{n / d}$ exactly when its cycles have length $d$ , equivalently when its step size has the form $r = (n / d) a$ with $a$ coprime to $d$ . Indeed, the order of the rotation by $r$ is the least positive $m$ with $m r \equiv 0 (m o d n)$ , and for $r = (n / d) a$ this least $m$ is exactly $d$ when $\gcd (a, d) = 1$ . Therefore the rotations of order $d$ are in bijection with the units $a \in (Z / d)^{\times}$ , so there are $φ (d)$ of them by [L1].

step 1.1L1algebra

3.1

Average the monomials over all $n$ rotations. Grouping them by the divisor $d$ from step 2.1 yields $Z (C_{n}) = \frac{1}{n} \sum_{d ∣ n} φ (d) s_{d}^{n / d}$ .

step 2.1∎

Depends on

Used by

Necklace count from the cyclic-group cycle index Corollary
The cycle index of C₄ Example
Two-colour necklaces of length 6 by Pólya Example
The cycle index of the dihedral group D₂ₙ Theorem

Dependency tree · two levels

13 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)