# Math Help - cycle decomposition 5

1. ## cycle decomposition 5

Show that a k-cycle has order k.

Let $f=(0,1,2,...,k-1)$ be the cycle. Then $f(n) = n$ if $n\not \in \{0,1,2,...,k-1\}$ and $f(m) = m+1(\bmod k)$ if $m\in \{0,1,2,...,k-1\}$. It is seems from this that $f^j (n) = n$ and $f^j(m) = m + j(\bmod k)$. For $f^j$ to be the identity we want $f^j (m) = m$, so $m+j(\bmod k) = m$. This happens if and only if $k|j$. The smallest such $j$ is therefore $k=j$.