On the number of invariant Sylow subgroups under coprime action -Antonio Beltrán, Changguo Shao

Question

I'm reading the papers of Antonio Beltrán, Changguo Shao. The article is On the number of invariant Sylow subgroups under coprime action:

https://www.researchgate.net/publication/318675516_On_the_number_of_invariant_Sylow_subgroups_under_coprime_action

In the proof of the theorem C, I see

I don't understand how $ν_p^A (H) | ν_p^A(K)$ and $ν_p^A (K) | ν_p^A(G)$.

Thank you very much.

Answer 1

That's the induction step. Firstly, observe that the pairs $K \& H$, as well as $G \& K$, satisfy the condition of the theorem. For $|G:H|=1$ the theorem holds. He then assumes that $m_0=|G:H|> 1$ and that the statement holds for every $m<|G:H|$ (in order to apply induction) and he proves it for $m_0 =|G:H|$.