Question

Express as a product of disjoint cycles:

$$\begin{gathered} \left(a\right)\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 2& 1& 3& 5& 4& 7& 9& 8& 6\end{array}\right)\left(b\right)\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 3& 5& 1& 2& 4& 6& 8& 9& 7\end{array}\right) \\ \left(c\right)\left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 3& 5& 1& 2& 4& 9& 8& 7& 6\end{array}\right)\left(d\right)\left(14\right)\left(27\right)\left(523\right)\left(34\right)\left(1472\right) \\ \left(e\right)\left(7236\right)\left(85\right)\left(571\right)\left(1537\right)\left(486\right) \end{gathered}$$

Short answer

The product of disjoint cycles is$\left(12\right)\left(45\right)\left(679\right)$

Step-by-step solution

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

We have thedefinition of cyclethat,

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.

Two cycles which have no elements in common are calleddisjoint cycles.

Now, the given permutation is,

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

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

We have to find the product of disjoint cycles.

To find the product of disjoint cycles

The above permutation can be expressed as product of disjoint cycles as,

$$\begin{gathered} \left(\begin{array}{ccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9\\ 2& 1& 3& 5& 4& 7& 9& 8& 6\end{array}\right)=\left(12\right)\left(3\right)\left(45\right)\left(679\right)\left(8\right) \\ =\left(12\right)\left(45\right)\left(679\right) \end{gathered}$$

Hence, the product of disjoint cycles is$\left(12\right)\left(45\right)\left(679\right)$.