Verify statement about conjugates in symmetric group At http://planetmath.org/simplicityofthealternatinggroups it states the following.


Let $\pi$ be a permutation
written as disjoint cycles
\[
\pi = (a_1, a_2, \ldots, a_k)(b_1, b_2, \ldots, b_l)\ldots
(c_1,\ldots, c_m)
\]
It is easy to check that for every permutation $\sigma \in S_n$ we
have
\[
\sigma \pi \sigma^{-1} = (\sigma(a_1), \sigma(a_2), \ldots, \sigma(a_k))\,
                         (\sigma(b_1),\sigma(b_2), \ldots
                         \sigma(b_l))\, \ldots (\sigma(c_1),\ldots,
                         \sigma(c_m)) 
\]


While it seems clear that this is true based on trying several concrete cases, I'm having trouble figuring out how I would check this. Some guidance on this would be much appreciated.

Applying on $\sigma(a_i)$ suppose $(\sigma \pi (\sigma)^{-1})\sigma(a_i)= \sigma \pi(a_i)=\sigma(a_{i+1})$. Now you are clear I think.
