Lagmental Vicfred

Conjugacy Classes in S_n Are Indexed by Partitions by Vicfred

tags: mathematics, permutation groups, group actions, conjugacy

Two permutations are conjugate exactly when their disjoint cycle decompositions have the same lengths. The point is to make the formal expression readable enough to audit line by line.

Start locally

A permutation in \(S_n\) is best read through its disjoint cycle type. Group actions then translate algebra into orbits \(Gx\), stabilisers \(G_x\), and fixed-point counts.

$$ \lambda=1^{m_1}2^{m_2}\cdots n^{m_n},\qquad\sum_iim_i=n $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ |\mathcal C_\lambda|=\frac{n!}{\prod_{i=1}^{n}i^{m_i}m_i!} $$

Compute before generalising

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \left|\mathcal C_{(3,2,1)}\subseteq S_6\right|=\frac{6!}{3^1\,1!\,2^1\,1!\,1^1\,1!}=120 $$

The global view

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad \lambda=1^{m_1}2^{m_2}\cdots n^{m_n},\qquad\sum_iim_i=n,\\[5pt] \mathsf{C}\;&:\quad |\mathcal C_\lambda|=\frac{n!}{\prod_{i=1}^{n}i^{m_i}m_i!}. \end{aligned} $$

Edge conditions

Cycle notation suppresses fixed points, so the ambient symmetric group still matters. The cycle \((1\,2\,3)\) in \(S_3\) and in \(S_8\) has different centralisers.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] |\mathcal C_\lambda|=\frac{n!}{\prod_{i=1}^{n}i^{m_i}m_i!} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sun 19 March 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.