Write a permutation of S_3 as the list [σ(1),σ(2),σ(3)]. If σ=[2,3,1] and τ=[2,1,3], what is στ (τ is applied first)?

Exercise · ZW-SYMMETRIC-GROUP-002 · ★☆☆ Basic

Write a permutation of \(S_3\) as the list \([\sigma(1),\sigma(2),\sigma(3)]\). If \(\sigma=[2,3,1]\) and \(\tau=[2,1,3]\), what is \(\sigma\tau\) (\(\tau\) is applied first)?

Choose one option.

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A. \([3,2,1]\)

Explanation

We have \(\sigma\tau(1)=\sigma(2)=3\), \(\sigma\tau(2)=\sigma(1)=2\) and \(\sigma\tau(3)=\sigma(3)=1\); the list is \([3,2,1]\). The list \([1,3,2]\) is \(\tau\sigma\), which is different.

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Review the concept: Symmetric group

The symmetric group \(S_n\) is the set of bijections \(\sigma\) of \(\{1,\dots,n\}\) onto itself, with composition \(\sigma\tau\) as the operation.

\[(\sigma\tau)(i)=\sigma(\tau(i))\]

Write a permutation of \(S_3\) as the list \([\sigma(1),\sigma(2),\sigma(3)]\). If \(\sigma=[2,3,1]\) and \(\tau=[2,1,3]\), what is \(\sigma\tau\) (\(\tau\) is applied first)?

Explanation

Review the concept: Symmetric group

More practice problems: symmetric group