End ExtendInvariantIrr.
Section Frobenius.
Variables (gT : finGroupType) (G K : {group gT}).
(* Because he only defines Frobenius groups in chapter 7, Isaacs does not     *)
(* state these theorems using the Frobenius property.                         *)
Hypothesis frobGK : [Frobenius G with kernel K].

(* Specifications: *)
(* This is Isaacs, Theorem 6.34(a1). *)
Theorem inertia_Frobenius_ker i : i != 0 -> 'I_G['chi[K]_i] = K.
(* This is Isaacs, Theorem 6.34(a2) *)
Theorem irr_induced_Frobenius_ker i : i != 0 -> 'Ind[G, K] 'chi_i \in irr G.
(* This is Isaacs, Theorem 6.34(b) *)
Theorem Frobenius_Ind_irrP j :
reflect (exists2 i, i != 0 & 'chi_j = 'Ind[G, K] 'chi_i)
(~~ (K \subset cfker 'chi_j)).