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