Let $σ∈S_n$ and $(1 2 3 4 ... k)∈S_n$ be a $k$-cycle. Show that $σ(1\,2\, 3\, 4\, ... k)σ^{−1}=(σ(1)\,σ(2)…σ(k))$, where $(σ(1)\,σ(2)…σ(k))$ is another k-cycle.
My current thought is: We know $σ$ and $σ^{−1}$ are $k$-cycles and $σ(σ^−1)=(1\, 2\, 3\, ... k)$. For $σ(1 \,2 \,3\,4\, ... k)σ^{−1}$, every element $n$ in $σ^{−1}$ first go to $n+1$ in $(1 \,2\,3\, 4\, ... k)$, and then go to $σ(n+1)$. So it just looks like $(σ(1)\,σ(2)\,…\,σ(k))$. Is this proof correct and rigorous enough?
