Section InternalAction.
Variables (pi : nat_pred) (gT : finGroupType).
Implicit Types G H K A X : {group gT}.

(* This is B & G, Proposition 1.5(c) *)
Proposition coprime_Hall_trans A G H1 H2 :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
pi.-Hall(G) H1 -> A \subset 'N(H1) ->
pi.-Hall(G) H2 -> A \subset 'N(H2) ->
exists2 x, x \in 'C_G(A) & H1 :=: H2 :^ x.