Question

Write each permutation in cycle notation:

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

Short answer

The cycle notation for the given permutation is$\left(173\right)$.

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 elements of set maps to themselves.

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

Now, the given permutation is:

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

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

Representation of cycle notation

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

$$\begin{gathered} \left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 7& 2& 1& 4& 5& 6& 3& 8& 9\end{array}\right)=\left(173\right)\left(2\right)\left(4\right)\left(5\right)\left(6\right)\left(8\right)\left(9\right) \\ =\left(173\right) \end{gathered}$$

Hence, the cycle notation for the given permutation is$\left(173\right)$.