Exercise 5 — A Complete Introduction to Abstract Algebra

Let $\sigma,\pi\in\mathrm{S}_{n}$ . Suppose $\sigma$ has cycle decomposition

\begin{pmatrix}a_{1}&a_{2}&\cdots&a_{k_{1}}\end{pmatrix}\begin{pmatrix}b_{1}&b_{2}&\cdots&b_{k_{2}}\end{pmatrix}\cdots,

where $a_{1},a_{2},\dots,a_{k_{1}},b_{1},b_{2},\dots,b_{k_{2}},\dots$ are all distinct. Show that

\pi\sigma\pi^{-1}=\begin{pmatrix}\pi(a_{1})&\cdots&\pi(a_{k_{1}})\end{pmatrix}\begin{pmatrix}\pi(b_{1})&\cdots&\pi(b_{k_{2}})\end{pmatrix}\cdots,

that is, $\pi\sigma\pi^{-1}$ is obtained from $\sigma$ by replacing each entry $i$ by $\pi(i)$ .

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