Question

Write each permutation in cycle notation:

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

Short answer

The cycle notation for the given permutation is$\left(1245789\right)$

Step-by-step solution

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

We have the definition of cyclethat,

Let $a_{1,}{a}_2,\dots ,a_k$be the distinct elements of the set role="math" localid="1651584419772" $\{1,2,3,\dots ,n\}$.

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 a cycle of length role="math" localid="1651584839790" $korkcycle$.

Now, the given permutation is,

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

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

Representation of cycle notation

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

$$\begin{gathered} \left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 2& 4& 3& 5& 7& 6& 8& 9& 1\end{array}\right)=\left(1245789\right)\left(3\right)\left(6\right) \\ =\left(1245789\right) \end{gathered}$$

Hence, the cycle notation for the given permutation is$\left(1245789\right)$.