theory Sylow
imports Coset Exponent
begin
text ‹See also @{cite "Kammueller-Paulson:1999"}.›
text ‹The combinatorial argument is in theory @{theory 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)"
by (simp add: coset_mult_assoc calM_def)
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: inj_on_def l_cancel [of m1 x y, THEN iffD1] H_def m1)
show "restrict (op ⊗ 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"
by (auto simp add: H_def)
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"
by (simp add: H_def)
lemma H_into_carrier_G: "x ∈ H ⟹ x ∈ carrier G"
by (simp add: H_def)
lemma in_H_imp_eq: "g ∈ H ⟹ M1 #> g = M1"
by (simp add: H_def)
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 ≠ {}"
apply (simp add: H_def)
apply (rule exI [of _ 𝟭])
apply simp
done
lemma H_is_subgroup: "subgroup H G"
apply (rule subgroupI)
apply (rule subsetI)
apply (erule H_into_carrier_G)
apply (rule H_not_empty)
apply (simp add: H_def)
apply clarify
apply (erule_tac P = "λz. lhs z = M1" for lhs in subst)
apply (simp add: coset_mult_assoc )
apply (blast intro: H_m_closed)
done
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 (simp add: card_cosets_equal rcosetsI)
lemma M1_RelM_rcosetGM1g: "g ∈ carrier G ⟹ (M1, M1 #> g) ∈ RelM"
apply (simp add: RelM_def calM_def card_M1)
apply (rule conjI)
apply (blast intro: rcosetGM1g_subset_G)
apply (simp add: card_M1 M1_cardeq_rcosetGM1g)
apply (metis M1_subset_G coset_mult_assoc coset_mult_one r_inv_ex)
done
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_imp_subset)
lemma inj_M_GmodH: "∃f ∈ M → rcosets H. inj_on f M"
apply (rule bexI)
apply (rule_tac [2] M_funcset_rcosets_H)
apply (rule inj_onI, simp)
apply (rule trans [OF _ M_elem_map_eq])
prefer 2 apply assumption
apply (rule M_elem_map_eq [symmetric, THEN trans], assumption)
apply (rule coset_mult_inv1)
apply (erule_tac [2] M_elem_map_carrier)+
apply (rule_tac [2] M1_subset_G)
apply (rule coset_join1 [THEN in_H_imp_eq])
apply (rule_tac [3] H_is_subgroup)
prefer 2 apply (blast intro: M_elem_map_carrier)
apply (simp add: coset_mult_inv2 H_def M_elem_map_carrier subset_eq)
done
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 #> (@g. g ∈ carrier G ∧ H #> g = C)) ∈ rcosets H → M"
apply (simp add: RCOSETS_def)
apply (fast intro: someI2
intro!: M1_in_M in_quotient_imp_closed [OF RelM_equiv M_in_quot _ M1_RelM_rcosetGM1g])
done
text ‹Close to a duplicate of ‹inj_M_GmodH›.›
lemma inj_GmodH_M: "∃g ∈ rcosets H→M. inj_on g (rcosets H)"
apply (rule bexI)
apply (rule_tac [2] rcosets_H_funcset_M)
apply (rule inj_onI)
apply (simp)
apply (rule trans [OF _ H_elem_map_eq])
prefer 2 apply assumption
apply (rule H_elem_map_eq [symmetric, THEN trans], assumption)
apply (rule coset_mult_inv1)
apply (erule_tac [2] H_elem_map_carrier)+
apply (rule_tac [2] H_is_subgroup [THEN subgroup.subset])
apply (rule coset_join2)
apply (blast intro: H_elem_map_carrier)
apply (rule H_is_subgroup)
apply (simp add: H_I coset_mult_inv2 H_elem_map_carrier)
done
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)"
apply (insert inj_M_GmodH inj_GmodH_M)
apply (blast intro: card_bij finite_M H_is_subgroup
rcosets_subset_PowG [THEN finite_subset]
finite_Pow_iff [THEN iffD2])
done
lemma index_lem: "card M * card H = order G"
by (simp add: cardMeqIndexH lagrange H_is_subgroup)
lemma lemma_leq1: "p^a ≤ card H"
apply (rule dvd_imp_le)
apply (rule div_combine [OF prime_imp_prime_elem[OF prime_p] not_dvd_M])
prefer 2 apply (blast intro: subgroup.finite_imp_card_positive H_is_subgroup)
apply (simp add: index_lem order_G power_add mult_dvd_mono multiplicity_dvd zero_less_m)
done
lemma lemma_leq2: "card H ≤ p^a"
apply (subst card_M1 [symmetric])
apply (cut_tac M1_inj_H)
apply (blast intro!: M1_subset_G intro: card_inj H_into_carrier_G finite_subset [OF _ finite_G])
done
lemma card_H_eq: "card H = p^a"
by (blast intro: le_antisym lemma_leq1 lemma_leq2)
end
lemma (in sylow) sylow_thm: "∃H. subgroup H G ∧ card H = p^a"
using lemma_A1
apply clarify
apply (frule existsM1inM, clarify)
apply (subgoal_tac "sylow_central G p a m M1 M")
apply (blast dest: sylow_central.H_is_subgroup sylow_central.card_H_eq)
apply (simp add: sylow_central_def sylow_central_axioms_def sylow_axioms calM_def RelM_def)
done
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"
by (simp add: sylow_def group_def)
subsection ‹Sylow's Theorem›
theorem sylow_thm:
"⟦prime p; group G; order G = (p^a) * m; finite (carrier G)⟧
⟹ ∃H. subgroup H G ∧ card H = p^a"
by (rule sylow.sylow_thm [of G p a m]) (simp add: sylow_eq sylow_axioms_def)
end