Equivalence Class of Fixed Element

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Let $\sigma \in S_n$.

Let $\mathcal R \paren \sigma$ denote the equivalence defined in Permutation Induces Equivalence Relation.

Let $i \in \N^*_{\le n}$.

Then:
 * $i \in \Fix \sigma$ $\eqclass i {\mathcal R \paren \sigma} = \set i$

where:
 * $\eqclass i {\mathcal R \paren \sigma}$ denotes the equivalence class of $i$ under $\mathcal R \paren \sigma$
 * $\Fix \sigma$ denotes the set of fixed elements of $\sigma$.

Proof
By the definition of an equivalence relation it is easily seen that $\set i \subseteq \eqclass i {\mathcal R \paren \sigma}$.

Suppose that $i \in \Fix \sigma$.

Let $j \in \eqclass i {\mathcal R \paren \sigma}$.

Then by Condition for Membership of Equivalence Class and Permutation Induces Equivalence Relation:
 * $j \in \eqclass i {\mathcal R \paren \sigma} \iff i \mathrel {\mathcal R_\sigma} j \implies \exists m \in \Z: \sigma^m \paren i = j$

And by Fixed Point of Permutation is Fixed Point of Power:
 * $\sigma^m \paren i = i \implies i = j$

Therefore:
 * $\eqclass i {\mathcal R \paren \sigma} \subseteq \set i$

and so:
 * $\eqclass i {\mathcal R \paren \sigma} = \set i$

For the converse, suppose that:
 * $\eqclass i {\mathcal R \paren \sigma} = \set i$.

It is seen that:
 * $i \mathrel {\mathcal R_\sigma} \sigma \paren i \iff \sigma \paren i \in \eqclass i {\mathcal R \paren \sigma}$

Therefore:
 * $\sigma \paren i = i \implies i \in \Fix \sigma$

Hence the result.