Library mathcomp.solvable.primitive_action

(* (c) Copyright 2006-2015 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B.                                  *)

Require Import mathcomp.ssreflect.ssreflect.

n-transitive and primitive actions: [primitive A, on S | to] <=> A acts on S in a primitive manner, i.e., A is transitive on S and A does not act on any nontrivial partition of S. imprimitivity_system A to S Q <=> Q is a non-trivial primitivity system for the action of A on S via to, i.e., Q is a non-trivial partiiton of S on which A acts. to * n == in the %act scope, the total action induced by the total action to on n.-tuples. via n_act to n. n.-dtuple S == the set of n-tuples with distinct values in S. [transitive^n A, on S | to] <=> A is n-transitive on S, i.e., A is transitive on n.-dtuple S == the set of n-tuples with distinct values in S.

Set Implicit Arguments.

Import GroupScope.

Section PrimitiveDef.

Variables (aT : finGroupType) (sT : finType).
Variables (A : {set aT}) (S : {set sT}) (to : {action aT &-> sT}).

Definition imprimitivity_system Q :=
[&& partition Q S, [acts A, on Q | to^*] & 1 < #|Q| < #|S|].

Definition primitive :=
[transitive A, on S | to] && ~~ [ Q, imprimitivity_system Q].

End PrimitiveDef.

Notation "[ 'primitive' A , 'on' S | to ]" := (primitive A S to)
(at level 0, format "[ 'primitive' A , 'on' S | to ]") : form_scope.

Section Primitive.

Variables (aT : finGroupType) (sT : finType).
Variables (G : {group aT}) (to : {action aT &-> sT}) (S : {set sT}).

Lemma trans_prim_astab x :
x \in S [transitive G, on S | to]
[primitive G, on S | to] = maximal_eq 'C_G[x | to] G.

Lemma prim_trans_norm (H : {group aT}) :
[primitive G, on S | to] H <| G
H \subset 'C_G(S | to) [transitive H, on S | to].

End Primitive.

Section NactionDef.

Variables (gT : finGroupType) (sT : finType).
Variables (to : {action gT &-> sT}) (n : nat).

Definition n_act (t : n.-tuple sT) a := [tuple of map (to^~ a) t].

Fact n_act_is_action : is_action setT n_act.

Canonical n_act_action := Action n_act_is_action.

End NactionDef.

Notation "to * n" := (n_act_action to n) : action_scope.

Section NTransitive.

Variables (gT : finGroupType) (sT : finType).
Variables (n : nat) (A : {set gT}) (S : {set sT}) (to : {action gT &-> sT}).

Definition dtuple_on := [set t : n.-tuple sT | uniq t & t \subset S].
Definition ntransitive := [transitive A, on dtuple_on | to × n].

Lemma dtuple_onP t :
reflect (injective (tnth t) i, tnth t i \in S) (t \in dtuple_on).

Lemma n_act_dtuple t a :
a \in 'N(S | to) t \in dtuple_on n_act to t a \in dtuple_on.

End NTransitive.

Implicit Arguments n_act [gT sT n].

Notation "n .-dtuple ( S )" := (dtuple_on n S)
(at level 8, format "n .-dtuple ( S )") : set_scope.

Notation "[ 'transitive' ^ n A , 'on' S | to ]" := (ntransitive n A S to)
(at level 0, n at level 8,
format "[ 'transitive' ^ n A , 'on' S | to ]") : form_scope.

Section NTransitveProp.

Variables (gT : finGroupType) (sT : finType).
Variables (to : {action gT &-> sT}) (G : {group gT}) (S : {set sT}).

Lemma card_uniq_tuple n (t : n.-tuple sT) : uniq t #|t| = n.

Lemma n_act0 (t : 0.-tuple sT) a : n_act to t a = [tuple].

Lemma dtuple_on_add n x (t : n.-tuple sT) :
([tuple of x :: t] \in n.+1.-dtuple(S)) =
[&& x \in S, x \notin t & t \in n.-dtuple(S)].

Lemma dtuple_on_add_D1 n x (t : n.-tuple sT) :
([tuple of x :: t] \in n.+1.-dtuple(S))
= (x \in S) && (t \in n.-dtuple(S :\ x)).

Lemma dtuple_on_subset n (S1 S2 : {set sT}) t :
S1 \subset S2 t \in n.-dtuple(S1) t \in n.-dtuple(S2).

Lemma n_act_add n x (t : n.-tuple sT) a :
n_act to [tuple of x :: t] a = [tuple of to x a :: n_act to t a].

Lemma ntransitive0 : [transitive^0 G, on S | to].

Lemma ntransitive_weak k m :
k m [transitive^m G, on S | to] [transitive^k G, on S | to].

Lemma ntransitive1 m :
0 < m [transitive^m G, on S | to] [transitive G, on S | to].

Lemma ntransitive_primitive m :
1 < m [transitive^m G, on S | to] [primitive G, on S | to].

End NTransitveProp.

Section NTransitveProp1.

Variables (gT : finGroupType) (sT : finType).
Variables (to : {action gT &-> sT}) (G : {group gT}) (S : {set sT}).

This is the forward implication of Aschbacher (15.12).1
This is the converse implication of Aschbacher (15.12).1