Section ExtremalClass.
Variables (gT : finGroupType) (G : {group gT}).
Inductive extremal_group_type :=
ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.
Definition index_extremal_group_type c :=
match c with
| ModularGroup => 0
| Dihedral => 1
| SemiDihedral => 2
| Quaternion => 3
| NotExtremal => 4
end%N.
Definition enum_extremal_groups :=
[:: ModularGroup; Dihedral; SemiDihedral; Quaternion].
End ExtremalClass.

(* Replacement for Section 4 proof. *)
Lemma odd_pgroup_rank1_cyclic gT p (G : {group gT}) :
p.-group G -> odd #|G| -> cyclic G = ('r_p(G) <= 1).