Exercise · ZW-SYMMETRIC-GROUP-010 · ★★★ Advanced
In \(S_4\) with \(\sigma=[2,3,4,1]\) and \(\tau=[1,2,4,3]\), which permutation \(x\) satisfies \(\sigma x=\tau\)?
Choose one option.
Show answer
- C. \([4,1,3,2]\)
Explanation
Multiplying by \(\sigma^{-1}\) on the left gives \(x=\sigma^{-1}\tau\), whose list is \([4,1,3,2]\). The option \(\tau\sigma^{-1}\) multiplies on the wrong side.
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.
\[(\sigma\tau)(i)=\sigma(\tau(i))\]
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