(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(*   This file establishes basic properties of several important classes of   *)
(* maximal subgroups: maximal, max and min normal, simple, characteristically *)
(* simple subgroups, the Frattini and Fitting subgroups, the Thompson         *)
(* critical subgroup, special and extra-special groups, and self-centralising *)
(* normal (SCN) subgroups. In detail, we define:                              *)
(*      charsimple G == G is characteristically simple (it has no nontrivial  *)
(*                      characteristic subgroups, and is nontrivial)          *)
(*           'Phi(G) == the Frattini subgroup of G, i.e., the intersection of *)
(*                      all its maximal proper subgroups.                     *)
(*             'F(G) == the Fitting subgroup of G, i.e., the largest normal   *)
(*                      nilpotent subgroup of G (defined as the (direct)      *)
(*                      product of all the p-cores of G).                     *)
(*      critical C G == C is a critical subgroup of G: C is characteristic    *)
(*                      (but not functorial) in G, the center of C contains   *)
(*                      both its Frattini subgroup and the commutator [G, C], *)
(*                      and is equal to the centraliser of C in G. The        *)
(*                      Thompson_critical theorem provides critical subgroups *)
(*                      for p-groups; we also show that in this case the      *)
(*                      centraliser of C in Aut G is a p-group as well.       *)
(*         special G == G is a special group: its center, Frattini, and       *)
(*                      derived sugroups coincide (we follow Aschbacher in    *)
(*                      not considering nontrivial elementary abelian groups  *)
(*                      as special); we show that a p-group factors under     *)
(*                      coprime action into special groups (Aschbacher 24.7). *)
(*    extraspecial G == G is a special group whose center has prime order     *)
(*                      (hence G is non-abelian).                             *)
(*           'SCN(G) == the set of self-centralising normal abelian subgroups *)
(*                      of G (the A <| G such that 'C_G(A) = A).              *)
(*         'SCN_n(G) == the subset of 'SCN(G) containing all groups with rank *)
(*                      at least n (i.e., A \in 'SCN(G) and 'm(A) >= n).      *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Defs.
Variable gT : finGroupType.
Implicit Types (A B D : {set gT}) (G : {group gT}).
Definition charsimple A := [min A of G | G :!=: 1 & G \char A].
Definition Frattini A := \bigcap_(G : {group gT} | maximal_eq G A) G.
Canonical Frattini_group A : {group gT} := Eval hnf in [group of Frattini A].
Definition Fitting A := \big[dprod/1]_(p <- primes #|A|) 'O_p(A).

Lemma Fitting_group_set G : group_set (Fitting G).
Canonical Fitting_group G := group (Fitting_group_set G).