# Theory Sylow

theory Sylow
imports Coset Exponent
```(*  Title:      HOL/Algebra/Sylow.thy
Author:     Florian Kammueller, with new proofs by L C Paulson
*)

theory Sylow
imports Coset Exponent
begin

text ‹The combinatorial argument is in theory @{text "Exponent"}.›

lemma le_extend_mult: "⟦0 < c; a ≤ b⟧ ⟹ a ≤ b * c"
for c :: nat
by (metis divisors_zero dvd_triv_left leI less_le_trans nat_dvd_not_less zero_less_iff_neq_zero)

locale sylow = group +
fixes p and a and m and calM and RelM
assumes prime_p: "prime p"
and order_G: "order G = (p^a) * m"
and finite_G[iff]: "finite (carrier G)"
defines "calM ≡ {s. s ⊆ carrier G ∧ card s = p^a}"
and "RelM ≡ {(N1, N2). N1 ∈ calM ∧ N2 ∈ calM ∧ (∃g ∈ carrier G. N1 = N2 #> g)}"
begin

lemma RelM_refl_on: "refl_on calM RelM"
by (auto simp: refl_on_def RelM_def calM_def) (blast intro!: coset_mult_one [symmetric])

lemma RelM_sym: "sym RelM"
proof (unfold sym_def RelM_def, clarify)
fix y g
assume "y ∈ calM"
and g: "g ∈ carrier G"
then have "y = y #> g #> (inv g)"
then show "∃g'∈carrier G. y = y #> g #> g'"
by (blast intro: g)
qed

lemma RelM_trans: "trans RelM"
by (auto simp add: trans_def RelM_def calM_def coset_mult_assoc)

lemma RelM_equiv: "equiv calM RelM"
unfolding equiv_def by (blast intro: RelM_refl_on RelM_sym RelM_trans)

lemma M_subset_calM_prep: "M' ∈ calM // RelM  ⟹ M' ⊆ calM"
unfolding RelM_def by (blast elim!: quotientE)

end

subsection ‹Main Part of the Proof›

locale sylow_central = sylow +
fixes H and M1 and M
assumes M_in_quot: "M ∈ calM // RelM"
and not_dvd_M: "¬ (p ^ Suc (multiplicity p m) dvd card M)"
and M1_in_M: "M1 ∈ M"
defines "H ≡ {g. g ∈ carrier G ∧ M1 #> g = M1}"
begin

lemma M_subset_calM: "M ⊆ calM"
by (rule M_in_quot [THEN M_subset_calM_prep])

lemma card_M1: "card M1 = p^a"
using M1_in_M M_subset_calM calM_def by blast

lemma exists_x_in_M1: "∃x. x ∈ M1"
using prime_p [THEN prime_gt_Suc_0_nat] card_M1
by (metis Suc_lessD card_eq_0_iff empty_subsetI equalityI gr_implies_not0 nat_zero_less_power_iff subsetI)

lemma M1_subset_G [simp]: "M1 ⊆ carrier G"
using M1_in_M M_subset_calM calM_def mem_Collect_eq subsetCE by blast

lemma M1_inj_H: "∃f ∈ H→M1. inj_on f H"
proof -
from exists_x_in_M1 obtain m1 where m1M: "m1 ∈ M1"..
have m1: "m1 ∈ carrier G"
by (simp add: m1M M1_subset_G [THEN subsetD])
show ?thesis
proof
show "inj_on (λz∈H. m1 ⊗ z) H"
by (simp add: H_def inj_on_def m1)
show "restrict ((⊗) m1) H ∈ H → M1"
proof (rule restrictI)
fix z
assume zH: "z ∈ H"
show "m1 ⊗ z ∈ M1"
proof -
from zH
have zG: "z ∈ carrier G" and M1zeq: "M1 #> z = M1"
show ?thesis
by (rule subst [OF M1zeq]) (simp add: m1M zG rcosI)
qed
qed
qed
qed

end

subsection ‹Discharging the Assumptions of ‹sylow_central››

context sylow
begin

lemma EmptyNotInEquivSet: "{} ∉ calM // RelM"
by (blast elim!: quotientE dest: RelM_equiv [THEN equiv_class_self])

lemma existsM1inM: "M ∈ calM // RelM ⟹ ∃M1. M1 ∈ M"
using RelM_equiv equiv_Eps_in by blast

lemma zero_less_o_G: "0 < order G"
by (simp add: order_def card_gt_0_iff carrier_not_empty)

lemma zero_less_m: "m > 0"
using zero_less_o_G by (simp add: order_G)

lemma card_calM: "card calM = (p^a) * m choose p^a"
by (simp add: calM_def n_subsets order_G [symmetric] order_def)

lemma zero_less_card_calM: "card calM > 0"
by (simp add: card_calM zero_less_binomial le_extend_mult zero_less_m)

lemma max_p_div_calM: "¬ (p ^ Suc (multiplicity p m) dvd card calM)"
proof
assume "p ^ Suc (multiplicity p m) dvd card calM"
with zero_less_card_calM prime_p
have "Suc (multiplicity p m) ≤ multiplicity p (card calM)"
by (intro multiplicity_geI) auto
then have "multiplicity p m < multiplicity p (card calM)" by simp
also have "multiplicity p m = multiplicity p (card calM)"
by (simp add: const_p_fac prime_p zero_less_m card_calM)
finally show False by simp
qed

lemma finite_calM: "finite calM"
unfolding calM_def by (rule finite_subset [where B = "Pow (carrier G)"]) auto

lemma lemma_A1: "∃M ∈ calM // RelM. ¬ (p ^ Suc (multiplicity p m) dvd card M)"
using RelM_equiv equiv_imp_dvd_card finite_calM max_p_div_calM by blast

end

subsubsection ‹Introduction and Destruct Rules for ‹H››

context sylow_central
begin

lemma H_I: "⟦g ∈ carrier G; M1 #> g = M1⟧ ⟹ g ∈ H"

lemma H_into_carrier_G: "x ∈ H ⟹ x ∈ carrier G"

lemma in_H_imp_eq: "g ∈ H ⟹ M1 #> g = M1"

lemma H_m_closed: "⟦x ∈ H; y ∈ H⟧ ⟹ x ⊗ y ∈ H"
by (simp add: H_def coset_mult_assoc [symmetric])

lemma H_not_empty: "H ≠ {}"
by (force simp add: H_def intro: exI [of _ 𝟭])

lemma H_is_subgroup: "subgroup H G"
proof (rule subgroupI)
show "H ⊆ carrier G"
using H_into_carrier_G by blast
show "⋀a. a ∈ H ⟹ inv a ∈ H"
by (metis H_I H_into_carrier_G H_m_closed M1_subset_G Units_eq Units_inv_closed Units_inv_inv coset_mult_inv1 in_H_imp_eq)
show "⋀a b. ⟦a ∈ H; b ∈ H⟧ ⟹ a ⊗ b ∈ H"
by (blast intro: H_m_closed)
qed (use H_not_empty in auto)

lemma rcosetGM1g_subset_G: "⟦g ∈ carrier G; x ∈ M1 #> g⟧ ⟹ x ∈ carrier G"
by (blast intro: M1_subset_G [THEN r_coset_subset_G, THEN subsetD])

lemma finite_M1: "finite M1"
by (rule finite_subset [OF M1_subset_G finite_G])

lemma finite_rcosetGM1g: "g ∈ carrier G ⟹ finite (M1 #> g)"
using rcosetGM1g_subset_G finite_G M1_subset_G cosets_finite rcosetsI by blast

lemma M1_cardeq_rcosetGM1g: "g ∈ carrier G ⟹ card (M1 #> g) = card M1"
by (metis M1_subset_G card_rcosets_equal rcosetsI)

lemma M1_RelM_rcosetGM1g:
assumes "g ∈ carrier G"
shows "(M1, M1 #> g) ∈ RelM"
proof -
have "M1 #> g ⊆ carrier G"
moreover have "card (M1 #> g) = p ^ a"
using assms by (simp add: card_M1 M1_cardeq_rcosetGM1g)
moreover have "∃h∈carrier G. M1 = M1 #> g #> h"
by (metis assms M1_subset_G coset_mult_assoc coset_mult_one r_inv_ex)
ultimately show ?thesis
by (simp add: RelM_def calM_def card_M1)
qed

end

subsection ‹Equal Cardinalities of ‹M› and the Set of Cosets›

text ‹Injections between @{term M} and @{term "rcosets⇘G⇙ H"} show that
their cardinalities are equal.›

lemma ElemClassEquiv: "⟦equiv A r; C ∈ A // r⟧ ⟹ ∀x ∈ C. ∀y ∈ C. (x, y) ∈ r"
unfolding equiv_def quotient_def sym_def trans_def by blast

context sylow_central
begin

lemma M_elem_map: "M2 ∈ M ⟹ ∃g. g ∈ carrier G ∧ M1 #> g = M2"
using M1_in_M M_in_quot [THEN RelM_equiv [THEN ElemClassEquiv]]
by (simp add: RelM_def) (blast dest!: bspec)

lemmas M_elem_map_carrier = M_elem_map [THEN someI_ex, THEN conjunct1]

lemmas M_elem_map_eq = M_elem_map [THEN someI_ex, THEN conjunct2]

lemma M_funcset_rcosets_H:
"(λx∈M. H #> (SOME g. g ∈ carrier G ∧ M1 #> g = x)) ∈ M → rcosets H"
by (metis (lifting) H_is_subgroup M_elem_map_carrier rcosetsI restrictI subgroup.subset)

lemma inj_M_GmodH: "∃f ∈ M → rcosets H. inj_on f M"
proof
let ?inv = "λx. SOME g. g ∈ carrier G ∧ M1 #> g = x"
show "inj_on (λx∈M. H #> ?inv x) M"
proof (rule inj_onI, simp)
fix x y
assume eq: "H #> ?inv x = H #> ?inv y" and xy: "x ∈ M" "y ∈ M"
have "x = M1 #> ?inv x"
by (simp add: M_elem_map_eq ‹x ∈ M›)
also have "... = M1 #> ?inv y"
proof (rule coset_mult_inv1 [OF in_H_imp_eq [OF coset_join1]])
show "H #> ?inv x ⊗ inv (?inv y) = H"
by (simp add: H_into_carrier_G M_elem_map_carrier xy coset_mult_inv2 eq subsetI)
qed (simp_all add: H_is_subgroup M_elem_map_carrier xy)
also have "... = y"
using M_elem_map_eq ‹y ∈ M› by simp
finally show "x=y" .
qed
show "(λx∈M. H #> ?inv x) ∈ M → rcosets H"
by (rule M_funcset_rcosets_H)
qed

end

subsubsection ‹The Opposite Injection›

context sylow_central
begin

lemma H_elem_map: "H1 ∈ rcosets H ⟹ ∃g. g ∈ carrier G ∧ H #> g = H1"
by (auto simp: RCOSETS_def)

lemmas H_elem_map_carrier = H_elem_map [THEN someI_ex, THEN conjunct1]

lemmas H_elem_map_eq = H_elem_map [THEN someI_ex, THEN conjunct2]

lemma rcosets_H_funcset_M:
"(λC ∈ rcosets H. M1 #> (SOME g. g ∈ carrier G ∧ H #> g = C)) ∈ rcosets H → M"
using in_quotient_imp_closed [OF RelM_equiv M_in_quot _  M1_RelM_rcosetGM1g]
by (simp add: M1_in_M H_elem_map_carrier RCOSETS_def)

lemma inj_GmodH_M: "∃g ∈ rcosets H→M. inj_on g (rcosets H)"
proof
let ?inv = "λx. SOME g. g ∈ carrier G ∧ H #> g = x"
show "inj_on (λC∈rcosets H. M1 #> ?inv C) (rcosets H)"
proof (rule inj_onI, simp)
fix x y
assume eq: "M1 #> ?inv x = M1 #> ?inv y" and xy: "x ∈ rcosets H" "y ∈ rcosets H"
have "x = H #> ?inv x"
by (simp add: H_elem_map_eq ‹x ∈ rcosets H›)
also have "... = H #> ?inv y"
proof (rule coset_mult_inv1 [OF coset_join2])
show "?inv x ⊗ inv (?inv y) ∈ carrier G"
by (simp add: H_elem_map_carrier ‹x ∈ rcosets H› ‹y ∈ rcosets H›)
then show "(?inv x) ⊗ inv (?inv y) ∈ H"
by (simp add: H_I H_elem_map_carrier xy coset_mult_inv2 eq)
show "H ⊆ carrier G"
qed (simp_all add: H_is_subgroup H_elem_map_carrier xy)
also have "... = y"
by (simp add: H_elem_map_eq ‹y ∈ rcosets H›)
finally show "x=y" .
qed
show "(λC∈rcosets H. M1 #> ?inv C) ∈ rcosets H → M"
using rcosets_H_funcset_M by blast
qed

lemma calM_subset_PowG: "calM ⊆ Pow (carrier G)"
by (auto simp: calM_def)

lemma finite_M: "finite M"
by (metis M_subset_calM finite_calM rev_finite_subset)

lemma cardMeqIndexH: "card M = card (rcosets H)"
using inj_M_GmodH inj_GmodH_M
by (blast intro: card_bij finite_M H_is_subgroup rcosets_subset_PowG [THEN finite_subset])

lemma index_lem: "card M * card H = order G"
by (simp add: cardMeqIndexH lagrange H_is_subgroup)

lemma card_H_eq: "card H = p^a"
proof (rule antisym)
show "p^a ≤ card H"
proof (rule dvd_imp_le)
show "p ^ a dvd card H"
apply (rule div_combine [OF prime_imp_prime_elem[OF prime_p] not_dvd_M])
show "0 < card H"
by (blast intro: subgroup.finite_imp_card_positive H_is_subgroup)
qed
next
show "card H ≤ p^a"
using M1_inj_H card_M1 card_inj finite_M1 by fastforce
qed

end

lemma (in sylow) sylow_thm: "∃H. subgroup H G ∧ card H = p^a"
proof -
obtain M where M: "M ∈ calM // RelM" "¬ (p ^ Suc (multiplicity p m) dvd card M)"
using lemma_A1 by blast
then obtain M1 where "M1 ∈ M"
by (metis existsM1inM)
define H where "H ≡ {g. g ∈ carrier G ∧ M1 #> g = M1}"
with M ‹M1 ∈ M›
interpret sylow_central G p a m calM RelM H M1 M
by unfold_locales (auto simp add: H_def calM_def RelM_def)
show ?thesis
using H_is_subgroup card_H_eq by blast
qed

text ‹Needed because the locale's automatic definition refers to
@{term "semigroup G"} and @{term "group_axioms G"} rather than
simply to @{term "group G"}.›
lemma sylow_eq: "sylow G p a m ⟷ group G ∧ sylow_axioms G p a m"