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This problem is taken from Hoffman kunz linear algebra section $5.3$ page no $155$

Let $\sigma$ and $\tau$ be the permutations of degree $4$ defined by

$$\sigma1= 2, \sigma2= 3,\sigma3= 4, \sigma3= 4$$

$$\tau1=3,\tau2=1,\tau3=2,\tau4=4$$

$(a)$ Is $\sigma$ odd or even? Is $\tau $ odd or even?

My attempt :$\sigma1= 2, \sigma2= 3,\sigma3= 4, \sigma3= 4 \implies \begin{pmatrix}1&2&3&4\\2&3&4&1\end{pmatrix}$

$$\sigma=(1234)$$

$$\sigma=\underbrace{(1234)}_{4 \text{number}}$$

$4$ is even $\implies \sigma$ is even

similarly $\tau=(132)$ ,$\tau=\underbrace{(132)}_{3 \text{number}}$ $3$ is odd implies $\tau$ is odd

Martin Sleziak
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jasmine
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    An odd length cycle is an even permutation. An even length cycle is an odd permutation. This follows from the definition of the parity of a permutation. Count the inverted pairs. Or write them as products of 2-cycles and count whether you need an even or an odd number of factors. Remembering that a 2-cycle is an odd permutation. – Jyrki Lahtonen Aug 06 '21 at 04:23

1 Answers1

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Well, the first issue here is that you need to look up the definition of odd and even permutations. Look through the answers on this question for that.

If you need help breaking up the permutations into transpositions look at the accepted answer here.

Can you take it from here?

Avatrin
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