Let

\(n\) be a positive integer and let

\(k\) be an integer in the range

\(2\leq k\leq n\text{.}\) A permutation

\(\pi\in S_n\) (see

Definition 2.1.1) is called a

\(k\)-cycle if there is a

\(k\)-element set

\(A=\{a_1,a_2,\ldots,a_k\}\subseteq \{1,2,\ldots,n\}\) such that

\(\pi(a_i)=a_{i+1}\) for

\(1\leq i\leq k-1\) and

\(\pi(a_k)=a_1\text{,}\) and

\(\pi(j)=j\) for

\(j\not\in
A\text{.}\) We use

cycle notation \((a_1a_2\cdots a_k)\) to denote the

\(k\)-cycle that acts as

\begin{equation*}
a_1{\to} a_2{\to} a_3{\to} \cdots{\to} a_k{\to} a_1
\end{equation*}

on the distinct positive integers \(a_1,a_2,\ldots,a_k\text{.}\) For example, the element \(\pi=[1,4,2,3]=(2,4,3)\) is a 3-cycle in \(S_4\) because \(\pi\) acts on the set \(A=\{2,3,4\}\) by

\begin{equation*}
2\to 4\to 3\to 2
\end{equation*}

and \(\pi\) acts on \(A^c=\{1\}\) as the identity. A \(1\)-cycle is defined to be the identity permutation, and may be written as \((a)\) in cycle notation, for any \(a\in \{1,2,\ldots,n\}\text{.}\) Note that cycle notation is not unique. For example, in \(S_4\) we have \((2,4,3)=(4,3,2)=(3,2,4)\) and \((1)=(2)=(3)=(4)\text{.}\) A permutation is called cyclic if it is a \(k\)-cycle for some \(k\text{,}\) \(1\leq k \leq
n\text{.}\) A 2-cycle is called a transposition.

Find all of the cyclic permutations in \(S_3\text{.}\) Find their inverses.

Find all of the cyclic permutations in \(S_4\text{.}\)