Section MinMaxGroup.
Variable gT : finGroupType.
Implicit Types gP : pred {group gT}.
Definition maxgroup A gP := maxset (fun A => group_set A && gP <<A>>%G) A.
Definition mingroup A gP := minset (fun A => group_set A && gP <<A>>%G) A.
Variable gP : pred {group gT}.
Arguments gP _%G.
Variable G : {group gT}.

Lemma mingroupP :
reflect (gP G /\ forall H, gP H -> H \subset G -> H :=: G) (mingroup G gP).