Question

Write each permutation in cycle notation:

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

Short answer

The cycle notation for given permutation is$\left(35798\right)$.

Step-by-step solution

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

We have the definition of cycle that,

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 element of set maps to themselves.

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

Now, the given permutation is:

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

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

Representation of cycle notation

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

$$\begin{gathered} \left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 1& 2& 5& 4& 7& 6& 9& 3& 8\end{array}\right)=\left(1\right)\left(2\right)\left(35798\right)\left(4\right)\left(6\right) \\ =\left(35798\right) \end{gathered}$$

Hence, the cycle notation for given permutation is$\left(35798\right)$.