I am reading an article about number of sylow subgroups. I have this theorem and don't understand some definitions which I am in italics below .
$\textbf{Theorem.}$ Suppose that $A$ is a finite group acting coprimely on a fnite group $G$, and let $H$ be an $A$-invariant subgroup of $G$. Let $C = \mathbb{C}_G(A)$. Then $\left[C:C\cap H\right]$ divides $\left[G:H\right]$.
Can you show me these definitions? I don't find this definition in any book. Maybe, I'm not knowledgeable enough to find it. Please help me. Thank you very much.