Section Sections.
Variables (gT : finGroupType).
Implicit Types (G : {group gT}) (s : section gT).
Canonical section_subType := Eval hnf in [newType for @pair_of_section gT].
Definition section_eqMixin := Eval hnf in [eqMixin of section gT by <:].
Canonical section_eqType := Eval hnf in EqType (section gT) section_eqMixin.
Definition section_choiceMixin := [choiceMixin of section gT by <:].
Canonical section_choiceType :=
Eval hnf in ChoiceType (section gT) section_choiceMixin.
Definition section_countMixin := [countMixin of section gT by <:].
Canonical section_countType :=
Eval hnf in CountType (section gT) section_countMixin.
Canonical section_subCountType := Eval hnf in [subCountType of section gT].
Definition section_finMixin := [finMixin of section gT by <:].
Canonical section_finType := Eval hnf in FinType (section gT) section_finMixin.
Canonical section_subFinType := Eval hnf in [subFinType of section gT].
Canonical section_group.
(* Isomorphic sections *)
Definition section_isog := [rel x y : section gT | x \isog y].
(* A witness of the isomorphism class of a section *)
Definition section_repr s := odflt (1 / 1)%sec (pick (section_isog ^~ s)).
Definition mksrepr G1 G2 := section_repr (mkSec G1 G2).

Lemma section_repr_isog s1 s2 :
s1 \isog s2 -> section_repr s1 = section_repr s2.