An equation with permutations, x^2 = sigma.

[Reid]

Homework Statement

x^2 = sigma.
The permutation sigma = (1 2 6 7 5 3 4), a cycle of length seven.

Determine x!

The Attempt at a Solution

I have tried a few times but my attempts are totally wrong. I don't know where to start! :S

I found a similar problem in the textbook. They could, by just lookin at the problem, say that there is no x for that (in the book not the one above) particular type of permutation. How? :S

 

[CompuChip]

I don't know if this is the best way, but it definitely works:
First determine what is the order of sigma. This can be used to solve the question. For example: suppose that the order is 3. Then sigma^3 = 1, so sigma^4 = sigma. From this, you can immediately find x.
If the order would be 4, on the other hand, then you would get sigma^4 = 1 so sigma^5 = sigma. The left hand side is not a square of anything, so there is no solution.

 

[Reid]

I'm not sure if i understood this correct. Let's see, with your reasoning...

I know that sigma^7=1, therefore sigma^8=sigma. Then x=sigma^4. Right?

 

[HallsofIvy]

Yes, that's exactly right. I didn't quite comprehend what CompuChip meant by "suppose that the order is 3" since the order was clearly 7! I'm glad you did.

 

[CompuChip]

I tried not to give away the answer completely. Didn't work