(* Title: HOL/Algebra/Sylow.thy
Author: Florian Kammueller, with new proofs by L C Paulson
*)
theory Sylow
imports Coset Exponent
begin
text \<open>See also @{cite "Kammueller-Paulson:1999"}.\<close>
text \<open>The combinatorial argument is in theory \<open>Exponent\<close>.\<close>
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 \<open>Main Part of the Proof\<close>
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 (\z\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"
by (auto simp add: H_def)
show ?thesis
by (rule subst [OF M1zeq]) (simp add: m1M zG rcosI)
qed
qed
qed
qed
end
subsection \<open>Discharging the Assumptions of \<open>sylow_central\<close>\<close>
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 \<open>Introduction and Destruct Rules for \<open>H\<close>\<close>
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 \ {}"
by (force simp add: H_def intro: exI [of _ \<one>])
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"
by (simp add: assms r_coset_subset_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 \<open>Equal Cardinalities of \<open>M\<close> and the Set of Cosets\<close>
text \<open>Injections between \<^term>\<open>M\<close> and \<^term>\<open>rcosets\<^bsub>G\<^esub> H\<close> show that
their cardinalities are equal.\<close>
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:
"(\x\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 = "\x. SOME g. g \ carrier G \ M1 #> g = x"
show "inj_on (\x\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 \<open>x \<in> M\<close>)
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 \<open>y \<in> M\<close> by simp
finally show "x=y" .
qed
show "(\x\M. H #> ?inv x) \ M \ rcosets H"
by (rule M_funcset_rcosets_H)
qed
end
subsubsection \<open>The Opposite Injection\<close>
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:
"(\C \ 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 = "\x. SOME g. g \ carrier G \ H #> g = x"
show "inj_on (\C\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 \<open>x \<in> rcosets H\<close>)
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 \<open>x \<in> rcosets H\<close> \<open>y \<in> rcosets H\<close>)
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"
by (simp add: H_is_subgroup subgroup.subset)
qed (simp_all add: H_is_subgroup H_elem_map_carrier xy)
also have "... = y"
by (simp add: H_elem_map_eq \<open>y \<in> rcosets H\<close>)
finally show "x=y" .
qed
show "(\C\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])
by (simp add: index_lem multiplicity_dvd order_G power_add)
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 \<open>M1 \<in> M\<close>
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 \<open>Needed because the locale's automatic definition refers to
\<^term>\<open>semigroup G\<close> and \<^term>\<open>group_axioms G\<close> rather than
simply to \<^term>\<open>group G\<close>.\<close>
lemma sylow_eq: "sylow G p a m \ group G \ sylow_axioms G p a m"
by (simp add: sylow_def group_def)
subsection \<open>Sylow's Theorem\<close>
theorem sylow_thm:
"\prime p; group G; order G = (p^a) * m; finite (carrier G)\
\<Longrightarrow> \<exists>H. subgroup H G \<and> card H = p^a"
by (rule sylow.sylow_thm [of G p a m]) (simp add: sylow_eq sylow_axioms_def)
end
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