Chapter 8

Prove that the set \(T\) of all the transpositions in \(S_{n}\) generates \(S_{n}\).

If \(\alpha=\left(a_{1} \cdots a_{2}\right)\) is a cycle of length \(s\), then \(\alpha^{s}=\varepsilon, \alpha^{2 \mathrm{~s}}=\varepsilon\), and \(\alpha^{3 s}=\varepsilon .\) Is \(\alpha^{k}=\varepsilon\) for any positive integer \(k

Let \(\alpha=\left(a_{1} \cdots a_{s}\right)\) and \(\beta=\left(b_{1} \cdots b_{s}\right)\) be cycles of the same length, and let \(\pi\) be any permutation. If \(\pi\left(a_{i}\right)=b_{i}\) for \(i=1, \ldots, s\), then \(\pi \alpha \pi^{-1}=\beta\). If \(\alpha\) is any cycle and \(\pi\) any permutation, \(\pi \alpha \pi^{-1}\) is called a conjugate of \(\alpha\).

Determine which of the following permutations is even, and which is odd. (a) \(\pi=\left(\begin{array}{llllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 7 & 4 & 1 & 5 & 6 & 2 & 3 & 8\end{array}\right)\) (b) \((71864)\) (c) \((12)(76)(345)\) (d) \((1276)(3241)(7812)\) (e) \((123)(2345)(1357)\)

Compute each of the following products in \(S_{9}\). (Write your answer as a single permutation.). (a) \((145)(37)(682)\) (b) \((17)(628)(9354)\) (c) \((71825)(36)(49)\) (d) \((12)(347)\) (e) \((147)(1678)(74132)\) \((f)(6148)(2345)(12493)\)

Prove that the set \(T_{1}=\\{(12),(13), \ldots,(1 n)\\}\) generates \(S_{n}\).

If \(\alpha=\left(a_{1} \cdots a_{5}\right)\) is any cycle of length \(s\), the order of \(\alpha\) is \(s\).

If \(\alpha\) and \(\beta\) are any two cycles of the same length \(s\), there is a permutation \(\pi \in S_{n}\) such that \(\beta=\pi \alpha \pi^{-1}\)

(a) The product of two even permutations is even. (b) The product of two odd permutations is even. (c) The product of an even permutation and an odd permutation is odd.

Write each of the following permutations in \(S_{9}\) as a product of disjoint cycles: (a) \(\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 4 & 9 & 2 & 5 & 1 & 7 & 6 & 8 & 3\end{array}\right)\) (b) \(\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 4 & 9 & 2 & 3 & 8 & 1 & 6 & 5\end{array}\right)\) (c) \(\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 9 & 5 & 3 & 1 & 2 & 4 & 8 & 6\end{array}\right)\) (d) \(\left(\begin{array}{ccccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 8 & 7 & 4 & 3 & 6 & 5 & 1 & 2\end{array}\right)\)

Prove that every even permutation is a product of one or more cycles of length \(3 .\) [HINT: \((13)(12)=(123) ;(12)(34)=(321)(134) \cdot]\) Conclude that the set \(U\) of all cycles of length 3 generates \(A_{s}\).

Find the order of each of the following permutatior (a) \((12)(345)\) (b) \((12)(3456)\) (c) \((1234)(567890)\)

Express each of the following as a product of transpositions in \(S_{8}\). (a) \((137428)\) (b) \((416)(8235)\) (c) \((123)(456)(1574)\) (d) \(\pi=\left(\begin{array}{llllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 1 & 4 & 2 & 8 & 7 & 6 & 5\end{array}\right)\)

Prove that the set \(U_{1}=\\{(123),(124), \ldots,(12 n)\\}\) generates \(A_{n} .\) [HINT: \((a b c)=\) \((1 c a)(1 a b),(1 a b)=(1 b 2)(12 a)(12 b)\), and \(\left.(1 b 2)=(12 b)^{2} \cdot\right]\)

The set of all the even permutations is a subgroup of \(S_{n}\). (It is denoted by \(A_{n}\) and is called the alternating group on \(n\) symbols.)

What is the order of \(\alpha \beta\), if \(\alpha\) and \(\beta\) are disjoint cycles of lengths 4 and 6 , pectively? (Explain why. Use the fact that disjoint cycles commute.)

If \(\alpha\) and \(\beta\) are disjoint cycles, then \(\pi \alpha \pi^{-1}\) and \(\pi \beta \pi^{-1}\) are disjoint cycles.

The pair of cycles \((12)\) and \((12 \cdots n)\) generates \(S_{n}\) \(\left[\mathrm{HINT}:(1 \cdots n)(12)(1 \cdots n)^{-1}=(23) ;(12)(23)(12)=(13)\right]\)

What is the order of \(\alpha \beta\), if \(\alpha\) and \(\beta\) are disjoint cycles of lengths \(r\) and \(s\), respecely. (Venture a guess, explain, but do not attempt a rigorous proof.)

Let \(\sigma\) be a product \(\alpha_{1} \cdots \alpha_{1}\) of \(t\) disjoint cycles of lengths \(l_{1}, \ldots, l_{t}\), respectively. Then \(\pi \sigma \pi^{-1}\) is also a product of \(t\) disjoint cycles of lengths \(l_{1}, \ldots, l_{t}\).

\text { Let } \alpha \text { and } \beta \text { be cycles (not necessarily disjoint). If } \alpha^{2}=\beta^{2} \text {, then } \alpha=\beta \text {. }

In \(S_{5}\), express each of the following as the square of a cycle (that is, express as \(\alpha^{2}\) where \(\alpha\) is a cycle): (a) \((132)\) (b) \((12345)\) (c) \((13)(24)\)