Question

Compute each product:

$\begin{array}{l}\left(a\right)\left(12\right)\left(23\right)\left(34\right)\left(b\right)\left(246\right)\left(147\right)\left(135\right)\\ \left(c\right)\left(12\right)\left(53214\right)\left(23\right)\left(d\right)\left(1234\right)\left(2345\right)\end{array}$

Short answer

The product is$\left(1356247\right)$.

Step-by-step solution

Step by Step Solution Step 1: To compute the permutation from the given cycles

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.

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

Now, the given cycle is,

$\left(246\right)\left(147\right)\left(135\right)$

We have to compute the product of these cycles to obtain the permutation.

Here, the given product is a composition of functions. So, the right-hand permutation is performed first. Thus, we will start mapping from 1 in the rightmost cycle.

$\left(246\right)\left(147\right)\left(135\right)=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 3& 4& 5& 7& 6& 2& 1\end{array}\right)$

Here, 1 is mapped to 3.

Next, 2 is mapped to 4.

Next, 3 is mapped to 5.

Next, 4 is mapped to 7.

Next, 5 is mapped to 1,1, which is mapped to 4 ,4, mapped to 6. So, 5 is mapped to 6. Next, 6 is mapped 2 and 7 is mapped to 1.

To obtain final product

From step 1, we get,

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

Here, 1 is mapped to 3, 3, which is mapped to 5, 5, mapped to 6, 6, which is mapped to 2, 2, mapped to 4, 4, which is mapped to 7 and 7 is mapped back to 1.

The final product is expressed in the cycle notation as,

$$\begin{gathered} \left(246\right)\left(147\right)\left(135\right)=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 3& 4& 5& 7& 6& 2& 1\end{array}\right) \\ =\left(1356247\right) \end{gathered}$$

Hence, the product is$\left(1356247\right)$.