I am reading Introduction to Abstract Algebra by Keith Nicholson and ran into a lemma that states:

If the identity permutation $\varepsilon$ can be written as a product of $n \geq 3$ transpositions, then it can be written as a product of $n - 2$ transpositions.

So let us look at $S_4$. So $\varepsilon = (1234) = (12)(23)(34)$ which is the product of 3 transpositions and thus satisfies the criteria for this lemma. So I then tried to write $\varepsilon$ as the product of 1 transposition and I did not see how this was possible. What am I missing here?



If you're using cycle notation in your post (which it seems you are), then the element $\alpha=(1234)$ is not the identity permutation ($\alpha$ has order $4$) - The identity permutation in $S_4$ is given by $\varepsilon=(1)(2)(3)(4)$.

  • 1
    $\begingroup$ Ahh, I see. Thank you. $\endgroup$
    – Ebearr
    Nov 10 '13 at 2:07

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