Let $G$ pe a $p$ group. I have to show that
Are there any theorem that can help me to prove this ? We have discussed the Sylow theorems but I don’t know how to apply them – if those are the theorems I need.
(Does this theorem also hold for infinite groups $G$ ?)
$G$ operates on the set of subgroups by conjugation.
What are the possible lengths of orbits? What does it mean if the orbit length is $1$?
Remark: We don’t use finiteness of $G$ here, but the subgroup counts involved should be finite for the statement to make sense.