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

AxiomOfChoice
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?

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

AxiomOfChoice
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.

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"