Question

Write each permutation in cycle notation:

$$\begin{gathered} \mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{}\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 7& 2& 1& 4& 5& 6& 3& 8& 9\end{array}\right)\mathbf{}\mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{}\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 2& 4& 3& 5& 7& 6& 8& 9& 1\end{array}\right) \\ \left(C\right)\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 4& 8& 1& 7& 5& 2& 6& 3& 9\end{array}\right)\mathbf{}\mathbf{\left(}\mathit{d}\mathbf{\right)}\mathbf{}\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 1& 2& 5& 4& 7& 6& 9& 3& 8\end{array}\right) \end{gathered}$$

Short answer

The cycle notation for the given permutation is$\left(1476283\right)$.

Step-by-step solution

Step by Step Solution Step 1: To obtain the cycle notation

We have the definition of the cycle,

Let $a_{1,}{a}_2,\dots ,a_k$be the distinct elements of the set $\{1,2,3,\dots ,n\}$.

Then, $(a_1{a}_2\dots a_k)$denotes the permutation in$S_n$that maps $a_1$to $a_2$, $a_2$to $a_3$, ..., $a_{k-1}$to $a_k$and $a_k$to $a_1$and every other elements of set maps to themselves.

$(a_1{a}_2\dots a_k)$is called a cycle of length k or k cycle.

Now, the given permutation is:

$\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 4& 8& 1& 7& 5& 2& 6& 3& 9\end{array}\right)$

Here, 1 is mapped to 4, 4, which is mapped to 7, 7, which is mapped to 6, 6, mapped to 2, 2, which is mapped to 8, 8, mapped to 3, and 3 is mapped back to 1. All other elements are mapped to themselves, that is, 5 is mapped to 5 and 9 is mapped to 9.

Representation of cycle notation

The above permutation can be represented in the cycle notation form as:

$$\begin{gathered} \left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 4& 8& 1& 7& 5& 2& 6& 3& 9\end{array}\right)=\left(1476283\right)\left(5\right)\left(9\right) \\ =\left(1476283\right) \end{gathered}$$

Hence, the cycle notation for the given permutation is$\left(1476283\right)$.