# If x is a cycle of length n, x^n is the identity.

Is it true that if $\sigma \in S_n$ is a cycle of length $k \leq n$, then $\sigma^k = \varepsilon$, where $\varepsilon$ is the identity permutation, and that $k$ is the least nonzero integer having this property?

## Answers and Replies

matt grime
Homework Helper
That is surely obvious, isn't it?

That is surely obvious, isn't it?
Not to me. Maybe I'm missing something small...if you can get me started on why it's the case, I can probably finish it out.

Hurkyl
Staff Emeritus
Gold Member
Special cases are always a good way to get started. Try k=1,2,3.

P.S. you meant "least positive integer"

matt grime
Homework Helper
A k-cycle has order k - it really is trivial. You only need to consider the case of

(123..k)

which just rotates the elements 1,..,k cyclically.

matt grime