Question

Find the order of each permutation.

(e)$\left(1234\right)\left(24\right)\left(43215\right)$

Short answer

The order of a permutation$\left(1234\right)\left(24\right)\left(43215\right)$ is 2.

Step-by-step solution

Order of a permutation

The order of a permutation $\sigma$ is the smallest positive integer $n$such that ${\sigma }^n=\in$, where$\in$ is the identity permutation.

Permutation σ2 and src="data:image/svg+xml;base64,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" role="math" localid="1651725581097" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/ab252c0c-4561-4f8f-9ded-b49b2396f88d.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220505%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220505T044403Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=c953bc101a9ad4b9465c257be6a60eca172f4944dd9c08d3ece6ccb95796a73c" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/ab252c0c-4561-4f8f-9ded-b49b2396f88d.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220505%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220505T044324Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=3e248c184c60622a96c10c4571ed2ecb93649c8198a431313f014bc338e60990" σ3

Apply the permutation $\sigma$twice. Then,

$$\begin{gathered} {\sigma }^2=\sigma ·\sigma \\ =\left(1234\right)\left(24\right)\left(43215\right)·\left(1234\right)\left(24\right)\left(43215\right) \\ =\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 3& 2& 1& 5& 4\end{array}\right).\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 3& 2& 1& 5& 4\end{array}\right) \\ =\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 1& 2& 3& 4& 5\end{array}\right) \end{gathered}$$

Here, $n=2$is the smallest positive integer such that${\sigma }^2=\in$

Therefore, the order of a permutation$\left(1234\right)\left(24\right)\left(43215\right)$ is 2.