Section Inertia.
Variable gT : finGroupType.
Definition inertia (B : {set gT}) (phi : 'CF(B)) :=
[set y in 'N(B) | (phi ^ y)%CF == phi].
Local Notation "''I[' phi ]" := (inertia phi) : group_scope.
Local Notation "''I_' G [ phi ]" := (G%g :&: 'I[phi]) : group_scope.
Fact group_set_inertia (H : {group gT}) phi : group_set 'I[phi : 'CF(H)].
Canonical inertia_group H phi := Group (@group_set_inertia H phi).
Local Notation "''I[' phi ]" := (inertia_group phi) : Group_scope.
Local Notation "''I_' G [ phi ]" := (G :&: 'I[phi])%G : Group_scope.
Variables G H : {group gT}.
Implicit Type phi : 'CF(H).

Lemma inertia_injective (chi : 'CF(H)) :
{in H &, injective chi} -> 'I[chi] = 'C(H).