(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
From mathcomp
From mathcomp
(******************************************************************************)
(*  Definition of Frobenius groups, some basic results, and the Frobenius     *)
(* theorem on the number of solutions of x ^+ n = 1.                          *)
(*    semiregular K H <->                                                     *)
(*       the internal action of H on K is semiregular, i.e., no nontrivial    *)
(*       elements of H and K commute; note that this is actually a symmetric  *)
(*       condition.                                                           *)
(*    semiprime K H <->                                                       *)
(*       the internal action of H on K is "prime", i.e., an element of K that *)
(*       centralises a nontrivial element of H must centralise all of H.      *)
(*    normedTI A G L <=>                                                      *)
(*       A is nonempty, strictly disjoint from its conjugates in G, and has   *)
(*       normaliser L in G.                                                   *)
(*    [Frobenius G = K ><| H] <=>                                             *)
(*       G is (isomorphic to) a Frobenius group with kernel K and complement  *)
(*       H. This is an effective predicate (in bool), which tests the         *)
(*       equality with the semidirect product, and then the fact that H is a  *)
(*       proper self-normalizing TI-subgroup of G.                            *)
(*    [Frobenius G with kernel H] <=>                                         *)
(*       G is (isomorphic to) a Frobenius group with kernel K; same as above, *)
(*       but without the semi-direct product.                                 *)
(*    [Frobenius G with complement H] <=>                                     *)
(*       G is (isomorphic to) a Frobenius group with complement H; same as    *)
(*       above, but without the semi-direct product. The proof that this form *)
(*       is equivalent to the above (i.e., the existence of Frobenius         *)
(*       kernels) requires character theory and will only be proved in the    *)
(*       vcharacter.v file.                                                   *)
(*    [Frobenius G] <=> G is a Frobenius group.                               *)
(*    Frobenius_action G H S to <->                                           *)
(*       The action to of G on S defines an isomorphism of G with a           *)
(*       (permutation) Frobenius group, i.e., to is faithful and transitive   *)
(*       on S, no nontrivial element of G fixes more than one point in S, and *)
(*       H is the stabilizer of some element of S, and non-trivial. Thus,     *)
(*        Frobenius_action G H S 'P                                           *)
(*       asserts that G is a Frobenius group in the classic sense.            *)
(*    has_Frobenius_action G H <->                                            *)
(*        Frobenius_action G H S to holds for some sT : finType, S : {set st} *)
(*        and to : {action gT &-> sT}. This is a predicate in Prop, but is    *)
(*        exactly reflected by [Frobenius G with complement H] : bool.        *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Definitions.
Variable gT : finGroupType.
Implicit Types A G K H L : {set gT}.
(* Corresponds to "H acts on K in a regular manner" in B & G. *)
Definition semiregular K H := {in H^#, forall x, 'C_K[x] = 1}.
(* Corresponds to "H acts on K in a prime manner" in B & G. *)
Definition semiprime K H := {in H^#, forall x, 'C_K[x] = 'C_K(H)}.
Definition normedTI A G L := [&& A != set0, trivIset (A :^: G) & 'N_G(A) == L].
Definition Frobenius_group_with_complement G H := (H != G) && normedTI H^# G H.
Definition Frobenius_group G :=
[exists H : {group gT}, Frobenius_group_with_complement G H].
Definition Frobenius_group_with_kernel_and_complement G K H :=
(K ><| H == G) && Frobenius_group_with_complement G H.
Definition Frobenius_group_with_kernel G K :=
[exists H : {group gT}, Frobenius_group_with_kernel_and_complement G K H].
Section FrobeniusAction.
Variables G H : {set gT}.
Variables (sT : finType) (S : {set sT}) (to : {action gT &-> sT}).
Definition Frobenius_action :=
[/\ [faithful G, on S | to],
[transitive G, on S | to],
{in G^#, forall x, #|'Fix_(S | to)[x]| <= 1},
H != 1
& exists2 u, u \in S & H = 'C_G[u | to]].
End FrobeniusAction.
Variant has_Frobenius_action G H : Prop :=
HasFrobeniusAction sT S to of @Frobenius_action G H sT S to.
End Definitions.
Arguments semiregular {gT} K%g H%g.
Arguments semiprime {gT} K%g H%g.
Arguments normedTI {gT} A%g G%g L%g.
Arguments Frobenius_group_with_complement {gT} G%g H%g.
Arguments Frobenius_group {gT} G%g.
Arguments Frobenius_group_with_kernel {gT} G%g K%g.
Arguments Frobenius_group_with_kernel_and_complement {gT} G%g K%g H%g.
Arguments Frobenius_action {gT} G%g H%g {sT} S%g to%act.
Arguments has_Frobenius_action {gT} G%g H%g.
Notation "[ 'Frobenius' G 'with' 'complement' H ]" :=
(Frobenius_group_with_complement G H)
(at level 0, G at level 50, H at level 35,
format "[ 'Frobenius'  G  'with'  'complement'  H ]") : group_scope.
Notation "[ 'Frobenius' G 'with' 'kernel' K ]" :=
(Frobenius_group_with_kernel G K)
(at level 0, G at level 50, K at level 35,
format "[ 'Frobenius'  G  'with'  'kernel'  K ]") : group_scope.
Notation "[ 'Frobenius' G ]" :=
(Frobenius_group G)
(at level 0, G at level 50,
format "[ 'Frobenius'  G ]") : group_scope.
Notation "[ 'Frobenius' G = K ><| H ]" :=
(Frobenius_group_with_kernel_and_complement G K H)
(at level 0, G at level 50, K, H at level 35,
format "[ 'Frobenius'  G  =  K  ><|  H ]") : group_scope.
Section FrobeniusBasics.
Variable gT : finGroupType.
Implicit Types (A B : {set gT}) (G H K L R X : {group gT}).

(* Specifications: *)
Lemma semiregular1l H : semiregular 1 H.
Lemma semiregularS K1 K2 A1 A2 :
K1 \subset K2 -> A1 \subset A2 -> semiregular K2 A2 -> semiregular K1 A1.
Lemma semiregular_prime H K : semiregular K H -> semiprime K H.
Lemma semiprime_regular H K : semiprime K H -> 'C_K(H) = 1 -> semiregular K H.
Lemma cent_semiprime H K X :
semiprime K H -> X \subset H -> X :!=: 1 -> 'C_K(X) = 'C_K(H).
Lemma stab_semiprime H K X :
semiprime K H -> X \subset K -> 'C_H(X) != 1 -> 'C_H(X) = H.
Lemma cent_semiregular H K X :
semiregular K H -> X \subset H -> X :!=: 1 -> 'C_K(X) = 1.
Lemma regular_norm_dvd_pred K H :
H \subset 'N(K) -> semiregular K H -> #|H| %| #|K|.-1.
Lemma regular_norm_coprime K H :
H \subset 'N(K) -> semiregular K H -> coprime #|K| #|H|.
Lemma semiregularJ K H x : semiregular K H -> semiregular (K :^ x) (H :^ x).
Lemma semiprimeJ K H x : semiprime K H -> semiprime (K :^ x) (H :^ x).
Lemma normedTI_P A G L :
reflect [/\ A != set0, L \subset 'N_G(A)
& {in G, forall g, ~~ [disjoint A & A :^ g] -> g \in L}]
(normedTI A G L).