Exercise 3.5.1 (Even and odd permutations)

Proof. We use

(check it!).

(Alternatively we may use $(a_1{a}_2\cdots a_m)=(a_1{a}_m)(a_1{a}_{m-1})\cdots (a_1{a}_2)$: see p. 107.)

(a)

(For Exercise 1.3.1.) In this Exercise, $$\begin{array}{llll}\hfill \sigma & =\left(\begin{array}{ccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right),& \hfill & \\ \hfill \tau & =\left(\begin{array}{ccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill 5\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 1\hfill \end{array}\right).& \hfill & \end{array}$$

By the solution of Ex.1.3.1,

$$\begin{array}{llll}\hfill \sigma & =(135)(24)=(13)(35)(24),& \hfill & \\ \hfill \tau & =(15)(23).& \hfill & \end{array}$$

So $\sigma$ is an odd permutation, and $\tau$ an even permutation.

(b)

(For Exercise 1.3.2.) Here $$\sigma =\left(\begin{array}{ccccccccccccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill & \hfill 10\hfill & \hfill 11\hfill & \hfill 12\hfill & \hfill 13\hfill & \hfill 14\hfill & \hfill 15\hfill \\ \hfill 13\hfill & \hfill 2\hfill & \hfill 15\hfill & \hfill 14\hfill & \hfill 10\hfill & \hfill 6\hfill & \hfill 12\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 7\hfill & \hfill 9\hfill & \hfill 5\hfill & \hfill 11\hfill & \hfill 8\hfill \end{array}\right),$$

$$\tau =\left(\begin{array}{ccccccccccccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill & \hfill 10\hfill & \hfill 11\hfill & \hfill 12\hfill & \hfill 13\hfill & \hfill 14\hfill & \hfill 15\hfill \\ \hfill 14\hfill & \hfill 9\hfill & \hfill 10\hfill & \hfill 2\hfill & \hfill 12\hfill & \hfill 6\hfill & \hfill 5\hfill & \hfill 11\hfill & \hfill 15\hfill & \hfill 3\hfill & \hfill 8\hfill & \hfill 7\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 13\hfill \end{array}\right).$$

Then

$$\begin{array}{llll}\hfill \sigma & =(113510)(3158)(414117129)& \hfill & \\ \hfill & =(113)(135)(510)(315)(158)(414)(1411)(117)(712)(129),& \hfill & \\ \hfill \tau & =(114)(2915134)(310)(5127)(811)& \hfill & \\ \hfill & =(114)(29)(915)(1513)(134)(310)(512)(127)(811),& \hfill & \\ \hfill {\sigma }^2& =(15)(1310)(3815)(41112)(7914)& \hfill & \\ \hfill & =(15)(1310)(38)(815)(411)(1112)(79)(914),& \hfill & \\ \hfill {\tau }^2& =(2154913)(5712)& \hfill & \\ \hfill & =(215)(154)(49)(913)(57)(712),& \hfill & \\ \hfill \mathit{𝜎𝜏}& =(1113)(24)(59871015)(1314)& \hfill & \\ \hfill & =(111)(113)(24)(59)(98)(87)(710)(1015)(1314),& \hfill & \\ \hfill \mathit{𝜏𝜎}& =(14)(29)(3131215115)(81014)& \hfill & \\ \hfill & =(14)(29)(313)(1312)(1215)(1511)(115)(810)(1014),& \hfill & \\ \hfill {\tau }^2\sigma & =(1215834141112137510)& \hfill & \\ \hfill & =(12)(215)(158)(83)(34)(414)(1411)(1112)(1213)(137)(75)(510).& \hfill & \end{array}$$

so $\sigma ,{\sigma }^2,{\tau }^2,{\tau }^2\sigma$ are even permutations and $\tau ,\mathit{𝜎𝜏},\mathit{𝜏𝜎}$ are odd permutations.