Section Nary.
Variables (I : finType) (P : pred I) (F : I -> {group gT}).
Canonical bigcap_group := group group_set_bigcap.
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 Lagrange_index G H K :
H \subset G -> K \subset H -> (#|G : H| * #|H : K|)%N = #|G : K|.