End Nary.
Canonical generated_group A : {group _} := Eval hnf in [group of <<A>>].
Canonical gcore_group G A : {group _} := Eval hnf in [group of gcore G A].
Canonical commutator_group A B : {group _} := Eval hnf in [group of [~: A, B]].
Canonical joing_group A B : {group _} := Eval hnf in [group of A <*> B].
Canonical cycle_group x : {group _} := Eval hnf in [group of <[x]>].
Definition joinG G H := joing_group G H.
Definition subgroups A := [set G : {group gT} | G \subset A].

Lemma prime_meetG G H : prime #|G| -> G :&: H != 1 -> G \subset H.
End Lagrange.