Question

Which of these permutations are even:

(c)$\left(12\right)\left(123\right)\left(1234\right)$

Short answer

The permutation$\left(12\right)\left(123\right)\left(1234\right)$ is an even permutation.

Step-by-step solution

Even permutation

A permutation is said to be even if it can be expressed as the product of an even number of transpositions.

Transpositions of Permutation (12)(123)(1234)

Express the permutation as the product of transpositions.

$$\begin{gathered} \left(12\right)\left(123\right)\left(1234\right)=\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 1& 3& 4\end{array}\right)\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 1& 4\end{array}\right)\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 2& 3& 4& 1& 5& 6\end{array}\right) \\ =\left(12\right)\left(12\right)\left(23\right)\left(12\right)\left(23\right)\left(34\right) \end{gathered}$$

Here, the permutation $\left(12\right)\left(123\right)\left(1234\right)$can be expressed as the product of 6 transpositions, which is an even transposition.

Therefore, the permutation$\left(12\right)\left(123\right)\left(1234\right)$ is an even permutation.