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?