<s>[INST] Prove that $ \sigma = \left( \begin{array}{ccccccccc} 1&2&3&4&5 \\ 2&3&4&1&5 \end{array} \right) \neq s_1 \circ s_2 \circ \ldots \circ s_k $ Prove that we can't write permutation $ \sigma = \left( \begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 1 & 5 \end{array} \right) $ as $s_1 \circ s_2 \circ \ldots \circ s_k $ where $s_1, s_2 , \ldots, s_k \in \left\{ \sigma_1= \left( \begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 1 & 4 & 5 \end{array} \right), \sigma_2 = \left( \begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 4 & 5 & 3 \end{array} \right)  \right\} $.
We have 
$ \sigma = \left( \begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 1 & 5 \end{array} \right) = (1234)
\\ \sigma_1 = \left( \begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 1 & 4 & 5 \end{array} \right) = (123)
\\ \sigma_2 =  \left( \begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 4 & 5 & 3 \end{array} \right) = (345) $
$ \mathrm{ord}( \sigma ) = 4 \\ \mathrm{ord} (\sigma_1) = 3 \\ \mathrm{ord}( \sigma_2) = 3$
Then I was thinking about $\mathrm{ord}(s_1 \circ \ldots \circ s_k)$ and maybe I should prove that $\mathrm{ord}(s_1 \circ \ldots \circ s_k) \neq 4 = \mathrm{ord}(\sigma)$ ?
But I have no idea how to prove it. Furthermore, I'm not sure, is it possible? Maybe is another way to do this task? Thanks in advance for any help.
 [/INST] Hint/Spoiler: Have you checked the parities of all the permutations here?
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