Permutation written as a single cycle

Question

How do I rewrite $(1\,2)(1\,3)(1\,4)(1\,5)$ as a single cycle? I have tried questions in the form: $(1\,4\,3\,5\,2)(4\,5\,3\,2\,1)$.

Answer 1

Keep in mind that as a composition of functions, $(1\,2)(1\,3)(1\,4)(1\,5)$ is to be read from right to left.

Now what happens to $1$? It's mapped to $5$ by the first transposition, and then all other transpositions fix $5$. So overall $$1\longrightarrow 5$$

What happens to $5$ now? The first transposition maps $5$ to $1$, then the second maps $1$ to $4$, and $4$ is fixed by the remaining transpositions. So we get: $$1\longrightarrow 5\longrightarrow 4$$ Similarly you get $ 4\longrightarrow 3$ and $3\longrightarrow 2$, so that in the end $$1\longrightarrow 5\longrightarrow 4\longrightarrow 3\longrightarrow 2$$ Which is usually written as $(1\,5\,4\,3\,2)$.

Answer 2

For a product of transpositions, just reverse the order! $$(1\,2)(1\,3)(1\,4)(1\,5)=(1\,5\,4\,3\,2)$$