# Library mathcomp.fingroup.fingroup

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B.                                  *)

From HB Require Import structures.
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice.
From mathcomp Require Import fintype div path tuple bigop prime finset.

Finite groups NB: See CONTRIBUTING.md for an introduction to HB concepts and commands. This file defines the main interface for finite groups: finGroupType == the structure for finite types with a group law The HB class is called FinGroup. {group gT} == type of groups with elements of type gT baseFinGroupType == the structure for finite types with a monoid law and an involutive antimorphism; finGroupType is derived from baseFinGroupType The HB class is called BaseFinGroup. FinGroupType mulVg == the finGroupType structure for an existing baseFinGroupType structure, built from a proof of the left inverse group axiom for that structure's operations [group of G] == a clone for an existing {group gT} structure on G : {set gT} (the existing structure might be for some delta-expansion of G) If gT implements finGroupType, then we can form {set gT}, the type of finite sets with elements of type gT (as finGroupType extends finType). The group law extends pointwise to {set gT}, which thus implements a sub- interface baseFinGroupType of finGroupType. To be consistent with the predType interface, this is done by coercion to FinGroup.arg_sort, an alias for FinGroup.sort. Accordingly, all pointwise group operations below have arguments of type (FinGroup.arg_sort) gT and return results of type FinGroup.sort gT. The notations below are declared in two scopes: group_scope (delimiter %g) for point operations and set constructs. Group_scope (delimiter %G) for explicit {group gT} structures. These scopes should not be opened globally, although group_scope is often opened locally in group-theory files (via Import GroupScope). As {group gT} is both a subtype and an interface structure for {set gT}, the fact that a given G : {set gT} is a group can (and usually should) be inferred by type inference with canonical structures. This means that all group' constructions (e.g., the normaliser 'N_G(H)) actually define sets with a canonical {group gT} structure; the %G delimiter can be used to specify the actual {group gT} structure (e.g., 'N_G(H)%G). Operations on elements of a group: x * y == the group product of x and y x ^+ n == the nth power of x, i.e., x * ... * x (n times) x^-1 == the group inverse of x x ^- n == the inverse of x ^+ n (notation for (x ^+ n)^-1) 1 == the unit element x ^ y == the conjugate of x by y (i.e., y^-1 * (x * y)) [~ x, y] == the commutator of x and y (i.e., x^-1 * x ^ y) [~ x1, ..., xn] == the commutator of x1, ..., xn (associating left) \prod_(i ...) x i == the product of the x i (order-sensitive) commute x y <-> x and y commute centralises x A <-> x centralises A 'C[x] == the set of elements that commute with x 'C_G[x] == the set of elements of G that commute with x < [x]> == the cyclic subgroup generated by the element x # [x] == the order of the element x, i.e., #|< [x]>| Operations on subsets/subgroups of a finite group: H * G == {xy | x \in H, y \in G} 1 or [1] or [1 gT] == the unit group [set: gT]%G == the group of all x : gT (in Group_scope) group_set G == G contains 1 and is closed under binary product; this is the characteristic property of the {group gT} subtype of {set gT} [subg G] == the subtype, set, or group of all x \in G: this notation is defined simultaneously in %type, %g and %G scopes, and G must denote a {group gT} structure (G is in the %G scope) subg, sgval == the projection into and injection from [subg G] H^# == the set H minus the unit element repr H == some element of H if 1 \notin H != set0, else 1 (repr is defined over sets of a baseFinGroupType, so it can be used, e.g., to pick right cosets.) x *: H == left coset of H by x lcosets H G == the set of the left cosets of H by elements of G H :* x == right coset of H by x rcosets H G == the set of the right cosets of H by elements of G #|G : H| == the index of H in G, i.e., #|rcosets G H| H :^ x == the conjugate of H by x x ^: H == the conjugate class of x in H classes G == the set of all conjugate classes of G G :^: H == {G :^ x | x \in H} class_support G H == {x ^ y | x \in G, y \in H} commg_set G H == { [~ x, y] | x \in G, y \in H}; NOT the commutator! <<H>> == the subgroup generated by the set H [~: G, H] == the commmutator subgroup of G and H, i.e., <<commg_set G H>>> [~: H1, ..., Hn] == commutator subgroup of H1, ..., Hn (left assoc.) H <*> G == the subgroup generated by sets H and G (H join G) (H * G)%G == the join of G H : {group gT} (convertible, but not identical to (G <*> H)%G) (\prod_(i ...) H i)%G == the group generated by the H i {in G, centralised H} <-> G centralises H {in G, normalised H} <-> G normalises H <-> forall x, x \in G -> H :^ x = H 'N(H) == the normaliser of H 'N_G(H) == the normaliser of H in G H <| G <=> H is a normal subgroup of G 'C(H) == the centraliser of H 'C_G(H) == the centraliser of H in G gcore H G == the largest subgroup of H normalised by G If H is a subgroup of G, this is the largest normal subgroup of G contained in H). abelian H <=> H is abelian subgroups G == the set of subgroups of G, i.e., the set of all H : {group gT} such that H \subset G In the notation below G is a variable that is bound in P. [max G | P] <=> G is the largest group such that P holds [max H of G | P] <=> H is the largest group G such that P holds [max G | P & Q] := [max G | P && Q], likewise [max H of G | P & Q] [min G | P] <=> G is the smallest group such that P holds [min G | P & Q] := [min G | P && Q], likewise [min H of G | P & Q] [min H of G | P] <=> H is the smallest group G such that P holds In addition to the generic suffixes described in ssrbool.v and finset.v, we associate the following suffixes to group operations: 1 - identity element, as in group1 : 1 \in G M - multiplication, as is invMg : (x * y)^-1 = y^-1 * x^-1 Also nat multiplication, for expgM : x ^+ (m * n) = x ^+ m ^+ n D - (nat) addition, for expgD : x ^+ (m + n) = x ^+ m * x ^+ n V - inverse, as in mulgV : x * x^-1 = 1 X - exponentiation, as in conjXg : (x ^+ n) ^ y = (x ^ y) ^+ n J - conjugation, as in orderJ : # [x ^ y] = # [x] R - commutator, as in conjRg : [~ x, y] ^ z = [~ x ^ z, y ^ z] Y - join, as in centY : 'C(G <*> H) = 'C(G) :&: 'C(H) We sometimes prefix these with an s' to indicate a set-lifted operation, e.g., conjsMg : (A * B) :^ x = A :^ x * B :^ x.

Set Implicit Arguments.

Declare Scope group_scope.
Declare Scope Group_scope.

Delimit Scope group_scope with g.
Delimit Scope Group_scope with G.

This module can be imported to open the scope for group element operations locally to a file, without exporting the Open to clients of that file (as Open would do).
Module GroupScope.
Open Scope group_scope.
End GroupScope.
Import GroupScope.

These are the operation notations introduced by this file.
Reserved Notation "[ ~ x1 , x2 , .. , xn ]" (at level 0,
format "'[ ' [ ~ x1 , '/' x2 , '/' .. , '/' xn ] ']'").
Reserved Notation "[ 1 gT ]" (at level 0, format "[ 1 gT ]").
Reserved Notation "[ 1 ]" (at level 0, format "[ 1 ]").
Reserved Notation "[ 'subg' G ]" (at level 0, format "[ 'subg' G ]").
Reserved Notation "A ^#" (at level 2, format "A ^#").
Reserved Notation "A :^ x" (at level 35, right associativity).
Reserved Notation "x ^: B" (at level 35, right associativity).
Reserved Notation "A :^: B" (at level 35, right associativity).
Reserved Notation "#| B : A |" (at level 0, B, A at level 99,
format "#| B : A |").
Reserved Notation "''N' ( A )" (at level 8, format "''N' ( A )").
Reserved Notation "''N_' G ( A )" (at level 8, G at level 2,
format "''N_' G ( A )").
Reserved Notation "A <| B" (at level 70, no associativity).
Reserved Notation "A <*> B" (at level 40, left associativity).
Reserved Notation "[ ~: A1 , A2 , .. , An ]" (at level 0,
format "[ ~: '[' A1 , '/' A2 , '/' .. , '/' An ']' ]").
Reserved Notation "[ 'max' A 'of' G | gP ]" (at level 0,
format "[ '[hv' 'max' A 'of' G '/ ' | gP ']' ]").
Reserved Notation "[ 'max' G | gP ]" (at level 0,
format "[ '[hv' 'max' G '/ ' | gP ']' ]").
Reserved Notation "[ 'max' A 'of' G | gP & gQ ]" (at level 0,
format "[ '[hv' 'max' A 'of' G '/ ' | gP '/ ' & gQ ']' ]").
Reserved Notation "[ 'max' G | gP & gQ ]" (at level 0,
format "[ '[hv' 'max' G '/ ' | gP '/ ' & gQ ']' ]").
Reserved Notation "[ 'min' A 'of' G | gP ]" (at level 0,
format "[ '[hv' 'min' A 'of' G '/ ' | gP ']' ]").
Reserved Notation "[ 'min' G | gP ]" (at level 0,
format "[ '[hv' 'min' G '/ ' | gP ']' ]").
Reserved Notation "[ 'min' A 'of' G | gP & gQ ]" (at level 0,
format "[ '[hv' 'min' A 'of' G '/ ' | gP '/ ' & gQ ']' ]").
Reserved Notation "[ 'min' G | gP & gQ ]" (at level 0,
format "[ '[hv' 'min' G '/ ' | gP '/ ' & gQ ']' ]").

We split the group axiomatisation in two. We define a class of "base groups", which are basically monoids with an involutive antimorphism, from which we derive the class of groups proper. This allows us to reuse much of the group notation and algebraic axioms for group subsets, by defining a base group class on them. We use class/mixins here rather than telescopes to be able to interoperate with the type coercions. Another potential benefit (not exploited here) would be to define a class for infinite groups, which could share all of the algebraic laws.

We want to use sort as a coercion class, both to infer argument scopes properly, and to allow groups and cosets to coerce to the base group of group subsets. However, the return type of group operations should NOT be a coercion class, since this would trump the real (head-normal) coercion class for concrete group types, thus spoiling the coercion of A * B to pred_sort in x \in A * B, or rho * tau to ffun and Funclass in (rho * tau) x, when rho tau : perm T. Therefore we define an alias of sort for argument types, and make it the default coercion FinGroup.base_type >-> Sortclass so that arguments of a functions whose parameters are of type, say, gT : finGroupType, can be coerced to the coercion class of arg_sort. Care should be taken, however, to declare the return type of functions and operators as FinGroup.sort gT rather than gT, e.g., mulg : gT -> gT -> FinGroup.sort gT. Note that since we do this here and in quotient.v for all the basic functions, the inferred return type should generally be correct.

#[arg_sort, short(type="baseFinGroupType")]
HB.structure Definition BaseFinGroup := { G of isMulBaseGroup G & Finite G }.

Module BaseFinGroupExports.
Bind Scope group_scope with BaseFinGroup.arg_sort.
Bind Scope group_scope with BaseFinGroup.sort.
Notation "[ 'baseFinGroupType' 'of' T ]" := (@BaseFinGroup.clone T%type _)
(at level 0, format "[ 'baseFinGroupType' 'of' T ]") : form_scope.
End BaseFinGroupExports.

Module Notations.
Section ElementOps.

Variable T : baseFinGroupType.
Notation rT := (BaseFinGroup.sort T).

Definition oneg : rT := Eval unfold oneg_subdef in @oneg_subdef T.
Definition mulg : T T rT := Eval unfold mulg_subdef in @mulg_subdef T.
Definition invg : T rT := Eval unfold invg_subdef in @invg_subdef T.
Definition expgn_rec (x : T) n : rT := iterop n mulg x oneg.

End ElementOps.

Definition expgn := nosimpl expgn_rec.

Notation "1" := (@oneg _) : group_scope.
Notation "x1 * x2" := (mulg x1 x2) : group_scope.
Notation "x ^-1" := (invg x) : group_scope.
Notation "x ^+ n" := (expgn x n) : group_scope.
Notation "x ^- n" := (x ^+ n)^-1 : group_scope.
End Notations.

#[short(type="finGroupType")]
HB.structure Definition FinGroup :=
{ G of BaseFinGroup_isGroup G & BaseFinGroup G }.

Module FinGroupExports.
Notation "[ 'finGroupType' 'of' T ]" := (@FinGroup.clone T%type _)
(at level 0, format "[ 'finGroupType' 'of' T ]") : form_scope.
Bind Scope group_scope with FinGroup.sort.
End FinGroupExports.

Notation "1" := oneg.
Infix "×" := mulg.
Notation "x ^-1" := (invg x).

Lemma mk_invgK : involutive invg.

Lemma mk_invMg : {morph invg : x y / x × y >-> y × x}.

#[compress_coercions]
HB.instance Definition _ (T : baseFinGroupType) :
Finite (BaseFinGroup.arg_sort T) := Finite.class T.

Arguments of conjg are restricted to true groups to avoid an improper interpretation of A ^ B with A and B sets, namely: {x^-1 * (y * z) | y \in A, x, z \in B}
Definition conjg (T : finGroupType) (x y : T) := y^-1 × (x × y).
Notation "x1 ^ x2" := (conjg x1 x2) : group_scope.

Definition commg (T : finGroupType) (x y : T) := x^-1 × x ^ y.
Notation "[ ~ x1 , x2 , .. , xn ]" := (commg .. (commg x1 x2) .. xn)
: group_scope.

Notation "\prod_ ( i <- r | P ) F" :=
(\big[mulg/1]_(i <- r | P%B) F%g) : group_scope.
Notation "\prod_ ( i <- r ) F" :=
(\big[mulg/1]_(i <- r) F%g) : group_scope.
Notation "\prod_ ( m <= i < n | P ) F" :=
(\big[mulg/1]_(m i < n | P%B) F%g) : group_scope.
Notation "\prod_ ( m <= i < n ) F" :=
(\big[mulg/1]_(m i < n) F%g) : group_scope.
Notation "\prod_ ( i | P ) F" :=
(\big[mulg/1]_(i | P%B) F%g) : group_scope.
Notation "\prod_ i F" :=
(\big[mulg/1]_i F%g) : group_scope.
Notation "\prod_ ( i : t | P ) F" :=
(\big[mulg/1]_(i : t | P%B) F%g) (only parsing) : group_scope.
Notation "\prod_ ( i : t ) F" :=
(\big[mulg/1]_(i : t) F%g) (only parsing) : group_scope.
Notation "\prod_ ( i < n | P ) F" :=
(\big[mulg/1]_(i < n | P%B) F%g) : group_scope.
Notation "\prod_ ( i < n ) F" :=
(\big[mulg/1]_(i < n) F%g) : group_scope.
Notation "\prod_ ( i 'in' A | P ) F" :=
(\big[mulg/1]_(i in A | P%B) F%g) : group_scope.
Notation "\prod_ ( i 'in' A ) F" :=
(\big[mulg/1]_(i in A) F%g) : group_scope.

Section PreGroupIdentities.

Variable T : baseFinGroupType.
Implicit Types x y z : T.

Lemma mulgA : associative mulgT.
Lemma mul1g : left_id 1 mulgT.
Lemma invgK : @involutive T invg.
Lemma invMg x y : (x × y)^-1 = y^-1 × x^-1.

Lemma invg_inj : @injective T T invg.

Lemma eq_invg_sym x y : (x^-1 == y) = (x == y^-1).

Lemma invg1 : 1^-1 = 1 :> T.

Lemma eq_invg1 x : (x^-1 == 1) = (x == 1).

Lemma mulg1 : right_id 1 mulgT.

Lemma expgnE x n : x ^+ n = expgn_rec x n.

Lemma expg0 x : x ^+ 0 = 1.
Lemma expg1 x : x ^+ 1 = x.

Lemma expgS x n : x ^+ n.+1 = x × x ^+ n.

Lemma expg1n n : 1 ^+ n = 1 :> T.

Lemma expgD x n m : x ^+ (n + m) = x ^+ n × x ^+ m.

Lemma expgSr x n : x ^+ n.+1 = x ^+ n × x.

Lemma expgM x n m : x ^+ (n × m) = x ^+ n ^+ m.

Lemma expgAC x m n : x ^+ m ^+ n = x ^+ n ^+ m.

Definition commute x y := x × y = y × x.

Lemma commute_refl x : commute x x.

Lemma commute_sym x y : commute x y commute y x.

Lemma commute1 x : commute x 1.

Lemma commuteM x y z : commute x y commute x z commute x (y × z).

Lemma commuteX x y n : commute x y commute x (y ^+ n).

Lemma commuteX2 x y m n : commute x y commute (x ^+ m) (y ^+ n).

Lemma expgVn x n : x^-1 ^+ n = x ^- n.

Lemma expgMn x y n : commute x y (x × y) ^+ n = x ^+ n × y ^+ n.

End PreGroupIdentities.

#[global] Hint Resolve commute1 : core.
Arguments invg_inj {T} [x1 x2].

Section GroupIdentities.

Variable T : finGroupType.
Implicit Types x y z : T.

Lemma mulVg : left_inverse 1 invg mulgT.

Lemma mulgV : right_inverse 1 invg mulgT.

Lemma mulKg : left_loop invg mulgT.

Lemma mulKVg : rev_left_loop invg mulgT.

Lemma mulgI : right_injective mulgT.

Lemma mulgK : right_loop invg mulgT.

Lemma mulgKV : rev_right_loop invg mulgT.

Lemma mulIg : left_injective mulgT.

Lemma eq_invg_mul x y : (x^-1 == y :> T) = (x × y == 1 :> T).

Lemma eq_mulgV1 x y : (x == y) = (x × y^-1 == 1 :> T).

Lemma eq_mulVg1 x y : (x == y) = (x^-1 × y == 1 :> T).

Lemma commuteV x y : commute x y commute x y^-1.

Lemma conjgE x y : x ^ y = y^-1 × (x × y).

Lemma conjgC x y : x × y = y × x ^ y.

Lemma conjgCV x y : x × y = y ^ x^-1 × x.

Lemma conjg1 x : x ^ 1 = x.

Lemma conj1g x : 1 ^ x = 1.

Lemma conjMg x y z : (x × y) ^ z = x ^ z × y ^ z.

Lemma conjgM x y z : x ^ (y × z) = (x ^ y) ^ z.

Lemma conjVg x y : x^-1 ^ y = (x ^ y)^-1.

Lemma conjJg x y z : (x ^ y) ^ z = (x ^ z) ^ y ^ z.

Lemma conjXg x y n : (x ^+ n) ^ y = (x ^ y) ^+ n.

Lemma conjgK : @right_loop T T invg conjg.

Lemma conjgKV : @rev_right_loop T T invg conjg.

Lemma conjg_inj : @left_injective T T T conjg.

Lemma conjg_eq1 x y : (x ^ y == 1) = (x == 1).

Lemma conjg_prod I r (P : pred I) F z :
(\prod_(i <- r | P i) F i) ^ z = \prod_(i <- r | P i) (F i ^ z).

Lemma commgEl x y : [~ x, y] = x^-1 × x ^ y.

Lemma commgEr x y : [~ x, y] = y^-1 ^ x × y.

Lemma commgC x y : x × y = y × x × [~ x, y].

Lemma commgCV x y : x × y = [~ x^-1, y^-1] × (y × x).

Lemma conjRg x y z : [~ x, y] ^ z = [~ x ^ z, y ^ z].

Lemma invg_comm x y : [~ x, y]^-1 = [~ y, x].

Lemma commgP x y : reflect (commute x y) ([~ x, y] == 1 :> T).

Lemma conjg_fixP x y : reflect (x ^ y = x) ([~ x, y] == 1 :> T).

Lemma commg1_sym x y : ([~ x, y] == 1 :> T) = ([~ y, x] == 1 :> T).

Lemma commg1 x : [~ x, 1] = 1.

Lemma comm1g x : [~ 1, x] = 1.

Lemma commgg x : [~ x, x] = 1.

Lemma commgXg x n : [~ x, x ^+ n] = 1.

Lemma commgVg x : [~ x, x^-1] = 1.

Lemma commgXVg x n : [~ x, x ^- n] = 1.

Other commg identities should slot in here.

End GroupIdentities.

#[global] Hint Rewrite mulg1 @mul1g invg1 @mulVg mulgV (@invgK) mulgK mulgKV
@invMg @mulgA : gsimpl.

Ltac gsimpl := autorewrite with gsimpl; try done.

Definition gsimp := (@mulg1, @mul1g, (@invg1, @invgK), (@mulgV, @mulVg)).
Definition gnorm := (gsimp, (@mulgK, @mulgKV, (@mulgA, @invMg))).

Arguments mulgI [T].
Arguments mulIg [T].
Arguments conjg_inj {T} x [x1 x2].
Arguments commgP {T x y}.
Arguments conjg_fixP {T x y}.

Section Repr.
Plucking a set representative.

Variable gT : baseFinGroupType.
Implicit Type A : {set gT}.

Definition repr A := if 1 \in A then 1 else odflt 1 [pick x in A].

Lemma mem_repr A x : x \in A repr A \in A.

Lemma card_mem_repr A : #|A| > 0 repr A \in A.

Lemma repr_set1 x : repr [set x] = x.

Lemma repr_set0 : repr set0 = 1.

End Repr.

Arguments mem_repr [gT A].

Section BaseSetMulDef.
We only assume a baseFinGroupType to allow this construct to be iterated.
Variable gT : baseFinGroupType.
Implicit Types A B : {set gT}.

Set-lifted group operations.

Definition set_mulg A B := mulg @2: (A, B).
Definition set_invg A := invg @^-1: A.

The pre-group structure of group subsets.
Time to open the bag of dirty tricks. When we define groups down below as a subtype of {set gT}, we need them to be able to coerce to sets in both set-style contexts (x \in G) and monoid-style contexts (G * H), and we need the coercion function to be EXACTLY the structure projection in BOTH cases -- otherwise the canonical unification breaks. Alas, Coq doesn't let us use the same coercion function twice, even when the targets are convertible. Our workaround (ab)uses the module system to declare two different identity coercions on an alias class.

Module GroupSet.
Definition sort (gT : baseFinGroupType) := {set gT}.
End GroupSet.
Identity Coercion GroupSet_of_sort : GroupSet.sort >-> set_of.

Module Type GroupSetBaseGroupSig.
Definition sort (gT : baseFinGroupType) := BaseFinGroup.arg_sort {set gT}.
End GroupSetBaseGroupSig.

Module MakeGroupSetBaseGroup (Gset_base : GroupSetBaseGroupSig).
Identity Coercion of_sort : Gset_base.sort >-> BaseFinGroup.arg_sort.
End MakeGroupSetBaseGroup.

Module Export GroupSetBaseGroup := MakeGroupSetBaseGroup GroupSet.

Section GroupSetMulDef.
Some of these constructs could be defined on a baseFinGroupType. We restrict them to proper finGroupType because we only develop the theory for that case.
Variable gT : finGroupType.
Implicit Types A B : {set gT}.
Implicit Type x y : gT.

Definition lcoset A x := mulg x @: A.
Definition rcoset A x := mulg^~ x @: A.
Definition lcosets A B := lcoset A @: B.
Definition rcosets A B := rcoset A @: B.
Definition indexg B A := #|rcosets A B|.

Definition conjugate A x := conjg^~ x @: A.
Definition conjugates A B := conjugate A @: B.
Definition class x B := conjg x @: B.
Definition classes A := class^~ A @: A.
Definition class_support A B := conjg @2: (A, B).

Definition commg_set A B := commg @2: (A, B).

These will only be used later, but are defined here so that we can keep all the Notation together.
"normalised" and "centralise[s|d]" are intended to be used with the {in ...} form, as in abelian below.
Definition normalised A := x, conjugate A x = A.
Definition centralises x A := y, y \in A commute x y.
Definition centralised A := x, centralises x A.

End GroupSetMulDef.

Arguments lcoset _ _%g _%g.
Arguments rcoset _ _%g _%g.
Arguments rcosets _ _%g _%g.
Arguments lcosets _ _%g _%g.
Arguments indexg _ _%g _%g.
Arguments conjugate _ _%g _%g.
Arguments conjugates _ _%g _%g.
Arguments class _ _%g _%g.
Arguments classes _ _%g.
Arguments class_support _ _%g _%g.
Arguments commg_set _ _%g _%g.
Arguments normaliser _ _%g.
Arguments centraliser _ _%g.
Arguments abelian _ _%g.
Arguments normal _ _%g _%g.
Arguments normalised _ _%g.
Arguments centralises _ _%g _%g.
Arguments centralised _ _%g.

Notation "[ 1 gT ]" := (1 : {set gT}) : group_scope.
Notation "[ 1 ]" := [1 FinGroup.sort _] : group_scope.

Notation "A ^#" := (A :\ 1) : group_scope.

Notation "x *: A" := ([set x%g] × A) : group_scope.
Notation "A :* x" := (A × [set x%g]) : group_scope.
Notation "A :^ x" := (conjugate A x) : group_scope.
Notation "x ^: B" := (class x B) : group_scope.
Notation "A :^: B" := (conjugates A B) : group_scope.

Notation "#| B : A |" := (indexg B A) : group_scope.

No notation for lcoset and rcoset, which are to be used mostly in curried form; x *: B and A :* 1 denote singleton products, so we can use mulgA, mulg1, etc, on, say, A :* 1 * B :* x. No notation for the set commutator generator set commg_set.

Notation "''N' ( A )" := (normaliser A) : group_scope.
Notation "''N_' G ( A )" := (G%g :&: 'N(A)) : group_scope.
Notation "A <| B" := (normal A B) : group_scope.
Notation "''C' ( A )" := (centraliser A) : group_scope.
Notation "''C_' G ( A )" := (G%g :&: 'C(A)) : group_scope.
Notation "''C_' ( G ) ( A )" := 'C_G(A) (only parsing) : group_scope.
Notation "''C' [ x ]" := 'N([set x%g]) : group_scope.
Notation "''C_' G [ x ]" := 'N_G([set x%g]) : group_scope.
Notation "''C_' ( G ) [ x ]" := 'C_G[x] (only parsing) : group_scope.

Section BaseSetMulProp.
Properties of the purely multiplicative structure.
Variable gT : baseFinGroupType.
Implicit Types A B C D : {set gT}.
Implicit Type x y z : gT.

Set product. We already have all the pregroup identities, so we only need to add the monotonicity rules.

Lemma mulsgP A B x :
reflect (imset2_spec mulg (mem A) (fun _mem B) x) (x \in A × B).

Lemma mem_mulg A B x y : x \in A y \in B x × y \in A × B.

Lemma prodsgP (I : finType) (P : pred I) (A : I {set gT}) x :
reflect (exists2 c, i, P i c i \in A i & x = \prod_(i | P i) c i)
(x \in \prod_(i | P i) A i).

Lemma mem_prodg (I : finType) (P : pred I) (A : I {set gT}) c :
( i, P i c i \in A i) \prod_(i | P i) c i \in \prod_(i | P i) A i.

Lemma mulSg A B C : A \subset B A × C \subset B × C.

Lemma mulgS A B C : B \subset C A × B \subset A × C.

Lemma mulgSS A B C D : A \subset B C \subset D A × C \subset B × D.

Lemma mulg_subl A B : 1 \in B A \subset A × B.

Lemma mulg_subr A B : 1 \in A B \subset A × B.

Lemma mulUg A B C : (A :|: B) × C = (A × C) :|: (B × C).

Lemma mulgU A B C : A × (B :|: C) = (A × B) :|: (A × C).

Set (pointwise) inverse.
Product with singletons.

Lemma set1gE : 1 = [set 1] :> {set gT}.
Lemma set1gP x : reflect (x = 1) (x \in [1 gT]).

Lemma mulg_set1 x y : [set x] :* y = [set x × y].

Lemma invg_set1 x : [set x]^-1 = [set x^-1].

End BaseSetMulProp.

Arguments set1gP {gT x}.
Arguments mulsgP {gT A B x}.
Arguments prodsgP {gT I P A x}.

Section GroupSetMulProp.
Constructs that need a finGroupType
Variable gT : finGroupType.
Implicit Types A B C D : {set gT}.
Implicit Type x y z : gT.

Left cosets.

Lemma lcosetE A x : lcoset A x = x *: A.

Lemma card_lcoset A x : #|x *: A| = #|A|.

Lemma mem_lcoset A x y : (y \in x *: A) = (x^-1 × y \in A).

Lemma lcosetP A x y : reflect (exists2 a, a \in A & y = x × a) (y \in x *: A).

Lemma lcosetsP A B C :
reflect (exists2 x, x \in B & C = x *: A) (C \in lcosets A B).

Lemma lcosetM A x y : (x × y) *: A = x *: (y *: A).

Lemma lcoset1 A : 1 *: A = A.

Lemma lcosetK : left_loop invg (fun x Ax *: A).

Lemma lcosetKV : rev_left_loop invg (fun x Ax *: A).

Lemma lcoset_inj : right_injective (fun x Ax *: A).

Lemma lcosetS x A B : (x *: A \subset x *: B) = (A \subset B).

Lemma sub_lcoset x A B : (A \subset x *: B) = (x^-1 *: A \subset B).

Lemma sub_lcosetV x A B : (A \subset x^-1 *: B) = (x *: A \subset B).

Right cosets.

Lemma rcosetE A x : rcoset A x = A :* x.

Lemma card_rcoset A x : #|A :* x| = #|A|.

Lemma mem_rcoset A x y : (y \in A :* x) = (y × x^-1 \in A).

Lemma rcosetP A x y : reflect (exists2 a, a \in A & y = a × x) (y \in A :* x).

Lemma rcosetsP A B C :
reflect (exists2 x, x \in B & C = A :* x) (C \in rcosets A B).

Lemma rcosetM A x y : A :* (x × y) = A :* x :* y.

Lemma rcoset1 A : A :* 1 = A.

Lemma rcosetK : right_loop invg (fun A xA :* x).

Lemma rcosetKV : rev_right_loop invg (fun A xA :* x).

Lemma rcoset_inj : left_injective (fun A xA :* x).

Lemma rcosetS x A B : (A :* x \subset B :* x) = (A \subset B).

Lemma sub_rcoset x A B : (A \subset B :* x) = (A :* x ^-1 \subset B).

Lemma sub_rcosetV x A B : (A \subset B :* x^-1) = (A :* x \subset B).

Inverse maps lcosets to rcosets
Conjugates.
Classes; not much for now.
Conjugate set.
Class support.
Groups (at last!)

Definition group_set A := (1 \in A) && (A × A \subset A).

Lemma group_setP A :
reflect (1 \in A {in A & A, x y, x × y \in A}) (group_set A).

Structure group_type : Type := Group {
gval :> GroupSet.sort gT;
_ : group_set gval
}.

Definition group_of : predArgType := group_type.
Identity Coercion type_of_group : group_of >-> group_type.

#[hnf] HB.instance Definition _ := [Finite of group_type by <:].

No predType or baseFinGroupType structures, as these would hide the group-to-set coercion and thus spoil unification.

Definition group (A : {set gT}) gA : groupT := @Group A gA.

Definition clone_group G :=
let: Group _ gP := G return {type of Group for G} groupT in fun kk gP.

Lemma group_inj : injective gval.
Lemma groupP (G : groupT) : group_set G.

Lemma congr_group (H K : groupT) : H = K H :=: K.

Lemma isgroupP A : reflect ( G : groupT, A = G) (group_set A).

Lemma group_set_one : group_set 1.

Canonical one_group := group group_set_one.
Canonical set1_group := @group [set 1] group_set_one.

Lemma group_setT : group_set (setTfor gT).

Canonical setT_group := group group_setT.

End GroupSetMulProp.

Arguments group_of gT%type.
Arguments lcosetP {gT A x y}.
Arguments lcosetsP {gT A B C}.
Arguments rcosetP {gT A x y}.
Arguments rcosetsP {gT A B C}.
Arguments group_setP {gT A}.
Arguments setT_group gT%type.

Notation "{ 'group' gT }" := (group_of gT)
(at level 0, format "{ 'group' gT }") : type_scope.

Notation "[ 'group' 'of' G ]" := (clone_group (@group _ G))
(at level 0, format "[ 'group' 'of' G ]") : form_scope.

Bind Scope Group_scope with group_type.
Bind Scope Group_scope with group_of.
Notation "1" := (one_group _) : Group_scope.
Notation "[ 1 gT ]" := (1%G : {group gT}) : Group_scope.
Notation "[ 'set' : gT ]" := (setT_group gT) : Group_scope.

These definitions come early so we can establish the Notation.
Canonical generated_unlockable := Unlockable generated.unlock.

Definition gcore (gT : finGroupType) (A B : {set gT}) := \bigcap_(x in B) A :^ x.
Definition joing (gT : finGroupType) (A B : {set gT}) := generated (A :|: B).
Definition commutator (gT : finGroupType) (A B : {set gT}) := generated (commg_set A B).
Definition cycle (gT : finGroupType) (x : gT) := generated [set x].
Definition order (gT : finGroupType) (x : gT) := #|cycle x|.

Arguments commutator _ _%g _%g.
Arguments joing _ _%g _%g.
Arguments generated _ _%g.

Helper notation for defining new groups that need a bespoke finGroupType. The actual group for such a type (say, my_gT) will be the full group, i.e., [set: my_gT] or [set: my_gT]%G, but Coq will not recognize specific notation for these because of the coercions inserted during type inference, unless they are defined as [set: gsort my_gT] using the Notation below.
Notation gsort gT := (BaseFinGroup.arg_sort gT%type) (only parsing).
Notation "<< A >>" := (generated A) : group_scope.
Notation "<[ x ] >" := (cycle x) : group_scope.
Notation "#[ x ]" := (order x) : group_scope.
Notation "A <*> B" := (joing A B) : group_scope.
Notation "[ ~: A1 , A2 , .. , An ]" :=
(commutator .. (commutator A1 A2) .. An) : group_scope.

Section GroupProp.

Variable gT : finGroupType.
Notation sT := {set gT}.
Implicit Types A B C D : sT.
Implicit Types x y z : gT.
Implicit Types G H K : {group gT}.

Section OneGroup.

Variable G : {group gT}.

Lemma valG : val G = G.

Non-triviality.

Lemma group1 : 1 \in G.
#[local] Hint Resolve group1 : core.

Lemma group1_contra x : x \notin G x != 1.

Lemma sub1G : [1 gT] \subset G.
Lemma subG1 : (G \subset [1]) = (G :==: 1).

Lemma setI1g : 1 :&: G = 1.
Lemma setIg1 : G :&: 1 = 1.

Lemma subG1_contra H : G \subset H G :!=: 1 H :!=: 1.

Lemma repr_group : repr G = 1.

Lemma cardG_gt0 : 0 < #|G|.

Lemma indexg_gt0 A : 0 < #|G : A|.

Lemma trivgP : reflect (G :=: 1) (G \subset [1]).

Lemma trivGP : reflect (G = 1%G) (G \subset [1]).

Lemma proper1G : ([1] \proper G) = (G :!=: 1).

Lemma in_one_group x : (x \in 1%G) = (x == 1).

Definition inE := (in_one_group, inE).

Lemma trivgPn : reflect (exists2 x, x \in G & x != 1) (G :!=: 1).

Lemma trivg_card_le1 : (G :==: 1) = (#|G| 1).

Lemma trivg_card1 : (G :==: 1) = (#|G| == 1%N).

Lemma cardG_gt1 : (#|G| > 1) = (G :!=: 1).

Lemma card_le1_trivg : #|G| 1 G :=: 1.

Lemma card1_trivg : #|G| = 1%N G :=: 1.

Inclusion and product.
Membership lemmas

Lemma groupM x y : x \in G y \in G x × y \in G.

Lemma groupX x n : x \in G x ^+ n \in G.

Lemma groupVr x : x \in G x^-1 \in G.

Lemma groupVl x : x^-1 \in G x \in G.

Lemma groupV x : (x^-1 \in G) = (x \in G).

Lemma groupMl x y : x \in G (x × y \in G) = (y \in G).

Lemma groupMr x y : x \in G (y × x \in G) = (y \in G).

Definition in_group := (group1, groupV, (groupMl, groupX)).

Lemma groupJ x y : x \in G y \in G x ^ y \in G.

Lemma groupJr x y : y \in G (x ^ y \in G) = (x \in G).

Lemma groupR x y : x \in G y \in G [~ x, y] \in G.

Lemma group_prod I r (P : pred I) F :
( i, P i F i \in G) \prod_(i <- r | P i) F i \in G.

Inverse is an anti-morphism.
Product idempotence
Left cosets

Lemma lcoset_refl x : x \in x *: G.

Lemma lcoset_sym x y : (x \in y *: G) = (y \in x *: G).

Lemma lcoset_eqP {x y} : reflect (x *: G = y *: G) (x \in y *: G).

Lemma lcoset_transl x y z : x \in y *: G (x \in z *: G) = (y \in z *: G).

Lemma lcoset_trans x y z : x \in y *: G y \in z *: G x \in z *: G.

Lemma lcoset_id x : x \in G x *: G = G.

Right cosets, with an elimination form for repr.

Lemma rcoset_refl x : x \in G :* x.

Lemma rcoset_sym x y : (x \in G :* y) = (y \in G :* x).

Lemma rcoset_eqP {x y} : reflect (G :* x = G :* y) (x \in G :* y).

Lemma rcoset_transl x y z : x \in G :* y (x \in G :* z) = (y \in G :* z).

Lemma rcoset_trans x y z : x \in G :* y y \in G :* z x \in G :* z.

Lemma rcoset_id x : x \in G G :* x = G.

Elimination form.
This form sometimes fails because ssreflect 1.1 delegates matching to the (weaker) primitive Coq algorithm for general (co)inductive type families.
Coset spaces.
Conjugates.
Classes
Subgroup Type construction. We only expect to use this for abstract groups, so we don't project the argument to a set.

Inductive subg_of : predArgType := Subg x & x \in G.
Definition sgval u := let: Subg x _ := u in x.
Definition subg_of_Sub := Eval hnf in [isSub for sgval].
#[hnf] HB.instance Definition _ := [Finite of subg_of by <:].

Lemma subgP u : sgval u \in G.
Lemma subg_inj : injective sgval.
Lemma congr_subg u v : u = v sgval u = sgval v.

Definition subg_one := Subg group1.
Definition subg_inv u := Subg (groupVr (subgP u)).
Definition subg_mul u v := Subg (groupM (subgP u) (subgP v)).
Lemma subg_oneP : left_id subg_one subg_mul.

Lemma subg_invP : left_inverse subg_one subg_inv subg_mul.
Lemma subg_mulP : associative subg_mul.

Lemma sgvalM : {in setT &, {morph sgval : x y / x × y}}.
Lemma valgM : {in setT &, {morph val : x y / (x : subg_of) × y >-> x × y}}.

Definition subg : gT subg_of := insubd (1 : subg_of).
Lemma subgK x : x \in G val (subg x) = x.
Lemma sgvalK : cancel sgval subg.
Lemma subg_default x : (x \in G) = false val (subg x) = 1.
Lemma subgM : {in G &, {morph subg : x y / x × y}}.

End OneGroup.

#[local] Hint Resolve group1 : core.

Lemma groupD1_inj G H : G^# = H^# G :=: H.

Lemma invMG G H : (G × H)^-1 = H × G.

Lemma mulSGid G H : H \subset G H × G = G.

Lemma mulGSid G H : H \subset G G × H = G.

Lemma mulGidPl G H : reflect (G × H = G) (H \subset G).

Lemma mulGidPr G H : reflect (G × H = H) (G \subset H).

Lemma comm_group_setP G H : reflect (commute G H) (group_set (G × H)).

Lemma card_lcosets G H : #|lcosets H G| = #|G : H|.

Group Modularity equations

Lemma group_modl A B G : A \subset G A × (B :&: G) = A × B :&: G.

Lemma group_modr A B G : B \subset G (G :&: A) × B = G :&: A × B.

End GroupProp.

#[global] Hint Extern 0 (is_true (1%g \in _)) ⇒ apply: group1 : core.
#[global] Hint Extern 0 (is_true (0 < #|_|)) ⇒ apply: cardG_gt0 : core.
#[global] Hint Extern 0 (is_true (0 < #|_ : _|)) ⇒ apply: indexg_gt0 : core.

Notation "G :^ x" := (conjG_group G x) : Group_scope.

Notation "[ 'subg' G ]" := (subg_of G) : type_scope.
Notation "[ 'subg' G ]" := [set: subg_of G] : group_scope.
Notation "[ 'subg' G ]" := [set: subg_of G]%G : Group_scope.

Bind Scope group_scope with subg_of.
Arguments subgK {gT G}.
Arguments sgvalK {gT G}.
Arguments subg_inj {gT G} [u1 u2] eq_u12 : rename.

Arguments trivgP {gT G}.
Arguments trivGP {gT G}.
Arguments lcoset_eqP {gT G x y}.
Arguments rcoset_eqP {gT G x y}.
Arguments mulGidPl {gT G H}.
Arguments mulGidPr {gT G H}.
Arguments comm_group_setP {gT G H}.
Arguments class_eqP {gT G x y}.
Arguments repr_classesP {gT G xG}.

Section GroupInter.

Variable gT : finGroupType.
Implicit Types A B : {set gT}.
Implicit Types G H : {group gT}.

Lemma group_setI G H : group_set (G :&: H).

Canonical setI_group G H := group (group_setI G H).

Section Nary.

Variables (I : finType) (P : pred I) (F : I {group gT}).

Lemma group_set_bigcap : group_set (\bigcap_(i | P i) F i).

Canonical bigcap_group := group group_set_bigcap.

End Nary.

Lemma group_set_generated (A : {set gT}) : group_set <<A>>.

Canonical generated_group A := group (group_set_generated 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 order_gt0 (x : gT) : 0 < #[x].

End GroupInter.

#[global] Hint Resolve order_gt0 : core.

Arguments generated_group _ _%g.
Arguments joing_group _ _%g _%g.
Arguments subgroups _ _%g.

Notation "G :&: H" := (setI_group G H) : Group_scope.
Notation "<< A >>" := (generated_group A) : Group_scope.
Notation "<[ x ] >" := (cycle_group x) : Group_scope.
Notation "[ ~: A1 , A2 , .. , An ]" :=
(commutator_group .. (commutator_group A1 A2) .. An) : Group_scope.
Notation "A <*> B" := (joing_group A B) : Group_scope.
Notation "G * H" := (joinG G H) : Group_scope.

Notation "\prod_ ( i <- r | P ) F" :=
(\big[joinG/1%G]_(i <- r | P%B) F%G) : Group_scope.
Notation "\prod_ ( i <- r ) F" :=
(\big[joinG/1%G]_(i <- r) F%G) : Group_scope.
Notation "\prod_ ( m <= i < n | P ) F" :=
(\big[joinG/1%G]_(m i < n | P%B) F%G) : Group_scope.
Notation "\prod_ ( m <= i < n ) F" :=
(\big[joinG/1%G]_(m i < n) F%G) : Group_scope.
Notation "\prod_ ( i | P ) F" :=
(\big[joinG/1%G]_(i | P%B) F%G) : Group_scope.
Notation "\prod_ i F" :=
(\big[joinG/1%G]_i F%G) : Group_scope.
Notation "\prod_ ( i : t | P ) F" :=
(\big[joinG/1%G]_(i : t | P%B) F%G) (only parsing) : Group_scope.
Notation "\prod_ ( i : t ) F" :=
(\big[joinG/1%G]_(i : t) F%G) (only parsing) : Group_scope.
Notation "\prod_ ( i < n | P ) F" :=
(\big[joinG/1%G]_(i < n | P%B) F%G) : Group_scope.
Notation "\prod_ ( i < n ) F" :=
(\big[joinG/1%G]_(i < n) F%G) : Group_scope.
Notation "\prod_ ( i 'in' A | P ) F" :=
(\big[joinG/1%G]_(i in A | P%B) F%G) : Group_scope.
Notation "\prod_ ( i 'in' A ) F" :=
(\big[joinG/1%G]_(i in A) F%G) : Group_scope.

Section Lagrange.

Variable gT : finGroupType.
Implicit Types G H K : {group gT}.

Lemma LagrangeI G H : (#|G :&: H| × #|G : H|)%N = #|G|.

Lemma divgI G H : #|G| %/ #|G :&: H| = #|G : H|.

Lemma divg_index G H : #|G| %/ #|G : H| = #|G :&: H|.

Lemma dvdn_indexg G H : #|G : H| %| #|G|.

Theorem Lagrange G H : H \subset G (#|H| × #|G : H|)%N = #|G|.

Lemma cardSg G H : H \subset G #|H| %| #|G|.

Lemma lognSg p G H : G \subset H logn p #|G| logn p #|H|.

Lemma piSg G H : G \subset H {subset \pi(gval G) \pi(gval H)}.

Lemma divgS G H : H \subset G #|G| %/ #|H| = #|G : H|.

Lemma divg_indexS G H : H \subset G #|G| %/ #|G : H| = #|H|.

Lemma coprimeSg G H p : H \subset G coprime #|G| p coprime #|H| p.

Lemma coprimegS G H p : H \subset G coprime p #|G| coprime p #|H|.

Lemma indexJg G H x : #|G :^ x : H :^ x| = #|G : H|.

Lemma indexgg G : #|G : G| = 1%N.

Lemma rcosets_id G : rcosets G G = [set G : {set gT}].

Lemma Lagrange_index G H K :
H \subset G K \subset H (#|G : H| × #|H : K|)%N = #|G : K|.

Lemma indexgI G H : #|G : G :&: H| = #|G : H|.

Lemma indexgS G H K : H \subset K #|G : K| %| #|G : H|.

Lemma indexSg G H K : H \subset K K \subset G #|K : H| %| #|G : H|.

Lemma indexg_eq1 G H : (#|G : H| == 1%N) = (G \subset H).

Lemma indexg_gt1 G H : (#|G : H| > 1) = ~~ (G \subset H).

Lemma index1g G H : H \subset G #|G : H| = 1%N H :=: G.

Lemma indexg1 G : #|G : 1| = #|G|.

Lemma indexMg G A : #|G × A : G| = #|A : G|.

Lemma rcosets_partition_mul G H : partition (rcosets H G) (H × G).

Lemma rcosets_partition G H : H \subset G partition (rcosets H G) G.

Lemma LagrangeMl G H : (#|G| × #|H : G|)%N = #|G × H|.

Lemma LagrangeMr G H : (#|G : H| × #|H|)%N = #|G × H|.

Lemma mul_cardG G H : (#|G| × #|H| = #|G × H|%g × #|G :&: H|)%N.

Lemma dvdn_cardMg G H : #|G × H| %| #|G| × #|H|.

Lemma cardMg_divn G H : #|G × H| = (#|G| × #|H|) %/ #|G :&: H|.

Lemma cardIg_divn G H : #|G :&: H| = (#|G| × #|H|) %/ #|G × H|.

Lemma TI_cardMg G H : G :&: H = 1 #|G × H| = (#|G| × #|H|)%N.

Lemma cardMg_TI G H : #|G| × #|H| #|G × H| G :&: H = 1.

Lemma coprime_TIg G H : coprime #|G| #|H| G :&: H = 1.

Lemma prime_TIg G H : prime #|G| ~~ (G \subset H) G :&: H = 1.

Lemma prime_meetG G H : prime #|G| G :&: H != 1 G \subset H.

Lemma coprime_cardMg G H : coprime #|G| #|H| #|G × H| = (#|G| × #|H|)%N.

Lemma coprime_index_mulG G H K :
H \subset G K \subset G coprime #|G : H| #|G : K| H × K = G.

End Lagrange.

Section GeneratedGroup.

Variable gT : finGroupType.
Implicit Types x y z : gT.
Implicit Types A B C D : {set gT}.
Implicit Types G H K : {group gT}.

Lemma subset_gen A : A \subset <<A>>.

Lemma sub_gen A B : A \subset B A \subset <<B>>.

Lemma mem_gen x A : x \in A x \in <<A>>.

Lemma generatedP x A : reflect ( G, A \subset G x \in G) (x \in <<A>>).

Lemma gen_subG A G : (<<A>> \subset G) = (A \subset G).

Lemma genGid G : <<G>> = G.

Lemma genGidG G : <<G>>%G = G.

Lemma gen_set_id A : group_set A <<A>> = A.

Lemma genS A B : A \subset B <<A>> \subset <<B>>.

Lemma gen0 : <<set0>> = 1 :> {set gT}.

Lemma gen_expgs A : {n | <<A>> = (1 |: A) ^+ n}.

Lemma gen_prodgP A x :
reflect ( n, exists2 c, i : 'I_n, c i \in A & x = \prod_i c i)
(x \in <<A>>).

Lemma genD A B : A \subset <<A :\: B>> <<A :\: B>> = <<A>>.

Lemma genV A : <<A^-1>> = <<A>>.

Lemma genJ A z : <<A :^z>> = <<A>> :^ z.

Lemma conjYg A B z : (A <*> B) :^z = A :^ z <*> B :^ z.

Lemma genD1 A x : x \in <<A :\ x>> <<A :\ x>> = <<A>>.

Lemma genD1id A : <<A^#>> = <<A>>.

Notation joingT := (@joing gT) (only parsing).
Notation joinGT := (@joinG gT) (only parsing).

Lemma joingE A B : A <*> B = <<A :|: B>>.

Lemma joinGE G H : (G × H)%G = (G <*> H)%G.

Lemma joingC : commutative joingT.

Lemma joing_idr A B : A <*> <<B>> = A <*> B.

Lemma joing_idl A B : <<A>> <*> B = A <*> B.

Lemma joing_subl A B : A \subset A <*> B.

Lemma joing_subr A B : B \subset A <*> B.

Lemma join_subG A B G : (A <*> B \subset G) = (A \subset G) && (B \subset G).

Lemma joing_idPl G A : reflect (G <*> A = G) (A \subset G).

Lemma joing_idPr A G : reflect (A <*> G = G) (A \subset G).

Lemma joing_subP A B G :
reflect (A \subset G B \subset G) (A <*> B \subset G).

Lemma joing_sub A B C : A <*> B = C A \subset C B \subset C.

Lemma genDU A B C : A \subset C <<C :\: A>> = <<B>> <<A :|: B>> = <<C>>.

Lemma joingA : associative joingT.

Lemma joing1G G : 1 <*> G = G.

Lemma joingG1 G : G <*> 1 = G.

Lemma genM_join G H : <<G × H>> = G <*> H.

Lemma mulG_subG G H K : (G × H \subset K) = (G \subset K) && (H \subset K).

Lemma mulGsubP K H G : reflect (K \subset G H \subset G) (K × H \subset G).

Lemma mulG_sub K H A : K × H = A K \subset A H \subset A.

Lemma trivMg G H : (G × H == 1) = (G :==: 1) && (H :==: 1).

Lemma comm_joingE G H : commute G H G <*> H = G × H.

Lemma joinGC : commutative joinGT.

Lemma joinGA : associative joinGT.

Lemma join1G : left_id 1%G joinGT.

Lemma joinG1 : right_id 1%G joinGT.

Lemma bigprodGEgen I r (P : pred I) (F : I {set gT}) :
(\prod_(i <- r | P i) <<F i>>)%G :=: << \bigcup_(i <- r | P i) F i >>.

Lemma bigprodGE I r (P : pred I) (F : I {group gT}) :
(\prod_(i <- r | P i) F i)%G :=: << \bigcup_(i <- r | P i) F i >>.

Lemma mem_commg A B x y : x \in A y \in B [~ x, y] \in [~: A, B].

Lemma commSg A B C : A \subset B [~: A, C] \subset [~: B, C].

Lemma commgS A B C : B \subset C [~: A, B] \subset [~: A, C].

Lemma commgSS A B C D :
A \subset B C \subset D [~: A, C] \subset [~: B, D].

Lemma der1_subG G : [~: G, G] \subset G.

Lemma comm_subG A B G : A \subset G B \subset G [~: A, B] \subset G.

Lemma commGC A B : [~: A, B] = [~: B, A].

Lemma conjsRg A B x : [~: A, B] :^ x = [~: A :^ x, B :^ x].

End GeneratedGroup.

Arguments gen_prodgP {gT A x}.
Arguments joing_idPl {gT G A}.
Arguments joing_idPr {gT A G}.
Arguments mulGsubP {gT K H G}.
Arguments joing_subP {gT A B G}.

Section Cycles.

Elementary properties of cycles and order, needed in perm.v. More advanced results on the structure of cyclic groups will be given in cyclic.v.

Variable gT : finGroupType.
Implicit Types x y : gT.
Implicit Types G : {group gT}.

Import Monoid.Theory.

Lemma cycle1 : <[1]> = [1 gT].

Lemma order1 : #[1 : gT] = 1%N.

Lemma cycle_id x : x \in <[x]>.

Lemma mem_cycle x i : x ^+ i \in <[x]>.

Lemma cycle_subG x G : (<[x]> \subset G) = (x \in G).

Lemma cycle_eq1 x : (<[x]> == 1) = (x == 1).

Lemma orderE x : #[x] = #|<[x]>|.

Lemma order_eq1 x : (#[x] == 1%N) = (x == 1).

Lemma order_gt1 x : (#[x] > 1) = (x != 1).

Lemma cycle_traject x : <[x]> =i traject (mulg x) 1 #[x].

Lemma cycle2g x : #[x] = 2 <[x]> = [set 1; x].

Lemma cyclePmin x y : y \in <[x]> {i | i < #[x] & y = x ^+ i}.

Lemma cycleP x y : reflect ( i, y = x ^+ i) (y \in <[x]>).

Lemma expg_order x : x ^+ #[x] = 1.

Lemma expg_mod p k x : x ^+ p = 1 x ^+ (k %% p) = x ^+ k.

Lemma expg_mod_order x i : x ^+ (i %% #[x]) = x ^+ i.

Lemma invg_expg x : x^-1 = x ^+ #[x].-1.

Lemma invg2id x : #[x] = 2 x^-1 = x.

Lemma cycleX x i : <[x ^+ i]> \subset <[x]>.

Lemma cycleV x : <[x^-1]> = <[x]>.

Lemma orderV x : #[x^-1] = #[x].

Lemma cycleJ x y : <[x ^ y]> = <[x]> :^ y.

Lemma orderJ x y : #[x ^ y] = #[x].

End Cycles.

Section Normaliser.

Variable gT : finGroupType.
Implicit Types x y z : gT.
Implicit Types A B C D : {set gT}.
Implicit Type G H K : {group gT}.

Lemma normP x A : reflect (A :^ x = A) (x \in 'N(A)).
Arguments normP {x A}.

Lemma group_set_normaliser A : group_set 'N(A).

Canonical normaliser_group A := group (group_set_normaliser A).

Lemma normsP A B : reflect {in A, normalised B} (A \subset 'N(B)).
Arguments normsP {A B}.

Lemma memJ_norm x y A : x \in 'N(A) (y ^ x \in A) = (y \in A).

Lemma norms_cycle x y : (<[y]> \subset 'N(<[x]>)) = (x ^ y \in <[x]>).

Lemma norm1 : '