[Math] Writing the identity permutation as a product of transpositions

Question

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?

Thanks.

Answer 1

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 $\varepsilon=(1)(2)(3)(4)$.