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: $(1\,4\,3\,5\,2)(4\,5\,3\,2\,1)$.

You can just individually check where each element $x \in {1,2,3,4,5 }$ maps and construct a single cycle — Theo C., Jul 07 '18 at 14:39
What are the steps to show that the answer is (12345)? Thanks for your quick response! — gradstudent3746, Jul 07 '18 at 14:42

score 1 · Accepted Answer · answered Jul 07 '18 at 14:48

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 $(1\,5\,4\,3\,2)$.

Thanks @ArnaudMortier for taking the time to explain the steps. Most helpful!! — gradstudent3746, Jul 07 '18 at 15:04

score 0 · Answer 2 · answered Jul 07 '18 at 14:39

0

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

answered Jul 07 '18 at 14:39

ə̷̶̸͇̘̜́̍͗̂̄︣͟

27,005

(15432) was my first answer on my first attempt, but I was informed the answer was (12345). Is this somehow a different form of the same answer? How would I show that it is (12345)? – gradstudent3746 Jul 07 '18 at 14:41
No, $(1,5,4,3,2)$ is not the same as $(1,2,3,4,5)$. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Jul 07 '18 at 14:42
I didnt think so, but I have now received two different responses that the answer is (12345) and someone else says (15432). – gradstudent3746 Jul 07 '18 at 14:43
Note that $$(1,2,3,4,5)=(1,5)(1,4)(1,3)(1,2)$$ which is something different. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Jul 07 '18 at 14:44
If it helps, please see the comments in the answer to this post on MSE. – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Jul 07 '18 at 14:44
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@gradstudent3746 The two answers you have gotten correspond to the two conventions for multiplying cycles. Some texts/people assume left to right, some right to left. Annoying, but that's how it is. – Ethan Bolker Jul 07 '18 at 14:48

Permutation written as a single cycle

2 Answers2