Question

Show that$\mathbf{\text{α}}\mathbf{}\mathbf{=}\mathbf{}\left(123\right)\mathbf{}\left(234\right)\mathbf{}\left(567\right)\mathbf{}\left(78910\right)$ has order 10 in$\left({\mathbf{\text{S}}}_{\mathbf{\text{n}}\mathbf{}}\mathbf{:}\mathbf{}\mathbf{10}\right)$.

Short answer

It is proved that $\alpha$does not have an order 10 in$\left(S_n\ge 10\right)$

Step-by-step solution

Required Theorem

Theorem7.25:The order of a permutation $\mathit{\tau }$ in${\mathit{S}}_{\mathbf{n}}$is the least common multiple of the length of the disjoint cycles whose product is$\mathit{\tau }$.

Proving that α = (123) (234) (567) (78910)   does not have order 10 in

It is given that,

$\alpha =\left(123\right)\left(234\right)\left(567\right)\left(78910\right)$

We can rewrite as the product of its disjoint cycles.

$$\begin{gathered} \alpha =\left(123\right)\left(234\right)\left(567\right)\left(78910\right) \\ =\left(12\right)\left(34\right)\left(5678910\right) \end{gathered}$$

Since the lengths of the disjoint cycle are 2, 2, 6, and (from the theorem), we know that the order should be the least common multiple of the length of an individual cycle.

As, LCM of (2,2,6) is 6. Therefore, the order of$\alpha$ is 6. Hence, it is proved that$\alpha$ does not have an order 10.