(* Title: HOL/Algebra/Sylow.thy ID: $Id: Sylow.thy,v 1.12 2007/10/23 21:27:23 nipkow Exp $ Author: Florian Kammueller, with new proofs by L C Paulson *) theory Sylow imports Coset begin section {* Sylow's Theorem *} text {* See also \cite{Kammueller-Paulson:1999}. *} text{*The combinatorial argument is in theory Exponent*} 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) )}" lemma (in sylow) RelM_refl: "refl calM RelM" apply (auto simp add: refl_def RelM_def calM_def) apply (blast intro!: coset_mult_one [symmetric]) done lemma (in sylow) RelM_sym: "sym RelM" proof (unfold sym_def RelM_def, clarify) fix y g assume "y ∈ calM" and g: "g ∈ carrier G" hence "y = y #> g #> (inv g)" by (simp add: coset_mult_assoc calM_def) thus "∃g'∈carrier G. y = y #> g #> g'" by (blast intro: g inv_closed) qed lemma (in sylow) RelM_trans: "trans RelM" by (auto simp add: trans_def RelM_def calM_def coset_mult_assoc) lemma (in sylow) RelM_equiv: "equiv calM RelM" apply (unfold equiv_def) apply (blast intro: RelM_refl RelM_sym RelM_trans) done lemma (in sylow) M_subset_calM_prep: "M' ∈ calM // RelM ==> M' ⊆ calM" apply (unfold RelM_def) apply (blast elim!: quotientE) done 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(exponent p m) dvd card(M))" and M1_in_M: "M1 ∈ M" defines "H == {g. g∈carrier G & M1 #> g = M1}" lemma (in sylow_central) M_subset_calM: "M ⊆ calM" by (rule M_in_quot [THEN M_subset_calM_prep]) lemma (in sylow_central) card_M1: "card(M1) = p^a" apply (cut_tac M_subset_calM M1_in_M) apply (simp add: calM_def, blast) done lemma card_nonempty: "0 < card(S) ==> S ≠ {}" by force lemma (in sylow_central) exists_x_in_M1: "∃x. x∈M1" apply (subgoal_tac "0 < card M1") apply (blast dest: card_nonempty) apply (cut_tac prime_p [THEN prime_imp_one_less]) apply (simp (no_asm_simp) add: card_M1) done lemma (in sylow_central) M1_subset_G [simp]: "M1 ⊆ carrier G" apply (rule subsetD [THEN PowD]) apply (rule_tac [2] M1_in_M) apply (rule M_subset_calM [THEN subset_trans]) apply (auto simp add: calM_def) done lemma (in sylow_central) M1_inj_H: "∃f ∈ H->M1. inj_on f H" proof - from exists_x_in_M1 obtain m1 where m1M: "m1 ∈ M1".. have m1G: "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 m1G) 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 subsection{*Discharging the Assumptions of @{text sylow_central}*} lemma (in sylow) EmptyNotInEquivSet: "{} ∉ calM // RelM" by (blast elim!: quotientE dest: RelM_equiv [THEN equiv_class_self]) lemma (in sylow) existsM1inM: "M ∈ calM // RelM ==> ∃M1. M1 ∈ M" apply (subgoal_tac "M ≠ {}") apply blast apply (cut_tac EmptyNotInEquivSet, blast) done lemma (in sylow) zero_less_o_G: "0 < order(G)" apply (unfold order_def) apply (blast intro: one_closed zero_less_card_empty) done lemma (in sylow) zero_less_m: "m > 0" apply (cut_tac zero_less_o_G) apply (simp add: order_G) done lemma (in sylow) card_calM: "card(calM) = (p^a) * m choose p^a" by (simp add: calM_def n_subsets order_G [symmetric] order_def) lemma (in sylow) zero_less_card_calM: "card calM > 0" by (simp add: card_calM zero_less_binomial le_extend_mult zero_less_m) lemma (in sylow) max_p_div_calM: "~ (p ^ Suc(exponent p m) dvd card(calM))" apply (subgoal_tac "exponent p m = exponent p (card calM) ") apply (cut_tac zero_less_card_calM prime_p) apply (force dest: power_Suc_exponent_Not_dvd) apply (simp add: card_calM zero_less_m [THEN const_p_fac]) done lemma (in sylow) finite_calM: "finite calM" apply (unfold calM_def) apply (rule_tac B = "Pow (carrier G) " in finite_subset) apply auto done lemma (in sylow) lemma_A1: "∃M ∈ calM // RelM. ~ (p ^ Suc(exponent p m) dvd card(M))" apply (rule max_p_div_calM [THEN contrapos_np]) apply (simp add: finite_calM equiv_imp_dvd_card [OF _ RelM_equiv]) done subsubsection{*Introduction and Destruct Rules for @{term H}*} lemma (in sylow_central) H_I: "[|g ∈ carrier G; M1 #> g = M1|] ==> g ∈ H" by (simp add: H_def) lemma (in sylow_central) H_into_carrier_G: "x ∈ H ==> x ∈ carrier G" by (simp add: H_def) lemma (in sylow_central) in_H_imp_eq: "g : H ==> M1 #> g = M1" by (simp add: H_def) lemma (in sylow_central) H_m_closed: "[| x∈H; y∈H|] ==> x ⊗ y ∈ H" apply (unfold H_def) apply (simp add: coset_mult_assoc [symmetric] m_closed) done lemma (in sylow_central) H_not_empty: "H ≠ {}" apply (simp add: H_def) apply (rule exI [of _ \<one>], simp) done lemma (in sylow_central) 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, clarify) apply (erule_tac P = "%z. ?lhs(z) = M1" in subst) apply (simp add: coset_mult_assoc ) apply (blast intro: H_m_closed) done lemma (in sylow_central) 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 (in sylow_central) finite_M1: "finite M1" by (rule finite_subset [OF M1_subset_G finite_G]) lemma (in sylow_central) finite_rcosetGM1g: "g∈carrier G ==> finite (M1 #> g)" apply (rule finite_subset) apply (rule subsetI) apply (erule rcosetGM1g_subset_G, assumption) apply (rule finite_G) done lemma (in sylow_central) M1_cardeq_rcosetGM1g: "g ∈ carrier G ==> card(M1 #> g) = card(M1)" by (simp (no_asm_simp) add: M1_subset_G card_cosets_equal rcosetsI) lemma (in sylow_central) M1_RelM_rcosetGM1g: "g ∈ carrier G ==> (M1, M1 #> g) ∈ RelM" apply (simp (no_asm) add: RelM_def calM_def card_M1 M1_subset_G) apply (rule conjI) apply (blast intro: rcosetGM1g_subset_G) apply (simp (no_asm_simp) add: card_M1 M1_cardeq_rcosetGM1g) apply (rule bexI [of _ "inv g"]) apply (simp_all add: coset_mult_assoc M1_subset_G) done subsection{*Equal Cardinalities of @{term M} and the Set of Cosets*} text{*Injections between @{term M} and @{term "rcosetsG H"} show that their cardinalities are equal.*} lemma ElemClassEquiv: "[| equiv A r; C ∈ A // r |] ==> ∀x ∈ C. ∀y ∈ C. (x,y)∈r" by (unfold equiv_def quotient_def sym_def trans_def, blast) lemma (in sylow_central) M_elem_map: "M2 ∈ M ==> ∃g. g ∈ carrier G & M1 #> g = M2" apply (cut_tac M1_in_M M_in_quot [THEN RelM_equiv [THEN ElemClassEquiv]]) apply (simp add: RelM_def) apply (blast dest!: bspec) done lemmas (in sylow_central) M_elem_map_carrier = M_elem_map [THEN someI_ex, THEN conjunct1] lemmas (in sylow_central) M_elem_map_eq = M_elem_map [THEN someI_ex, THEN conjunct2] lemma (in sylow_central) M_funcset_rcosets_H: "(%x:M. H #> (SOME g. g ∈ carrier G & M1 #> g = x)) ∈ M -> rcosets H" apply (rule rcosetsI [THEN restrictI]) apply (rule H_is_subgroup [THEN subgroup.subset]) apply (erule M_elem_map_carrier) done lemma (in sylow_central) 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_closed M_elem_map_carrier inv_closed) apply (simp add: coset_mult_inv2 H_def M_elem_map_carrier subset_def) done subsubsection{*The Opposite Injection*} lemma (in sylow_central) H_elem_map: "H1 ∈ rcosets H ==> ∃g. g ∈ carrier G & H #> g = H1" by (auto simp add: RCOSETS_def) lemmas (in sylow_central) H_elem_map_carrier = H_elem_map [THEN someI_ex, THEN conjunct1] lemmas (in sylow_central) H_elem_map_eq = H_elem_map [THEN someI_ex, THEN conjunct2] lemma EquivElemClass: "[|equiv A r; M ∈ A//r; M1∈M; (M1,M2) ∈ r |] ==> M2 ∈ M" by (unfold equiv_def quotient_def sym_def trans_def, blast) lemma (in sylow_central) 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!: restrictI M1_in_M EquivElemClass [OF RelM_equiv M_in_quot _ M1_RelM_rcosetGM1g]) done text{*close to a duplicate of @{text inj_M_GmodH}*} lemma (in sylow_central) 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: m_closed inv_closed H_elem_map_carrier) apply (rule H_is_subgroup) apply (simp add: H_I coset_mult_inv2 M1_subset_G H_elem_map_carrier) done lemma (in sylow_central) calM_subset_PowG: "calM ⊆ Pow(carrier G)" by (auto simp add: calM_def) lemma (in sylow_central) finite_M: "finite M" apply (rule finite_subset) apply (rule M_subset_calM [THEN subset_trans]) apply (rule calM_subset_PowG, blast) done lemma (in sylow_central) 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 (in sylow_central) index_lem: "card(M) * card(H) = order(G)" by (simp add: cardMeqIndexH lagrange H_is_subgroup) lemma (in sylow_central) lemma_leq1: "p^a ≤ card(H)" apply (rule dvd_imp_le) apply (rule div_combine [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 power_exponent_dvd zero_less_m) done lemma (in sylow_central) 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 (in sylow_central) card_H_eq: "card(H) = p^a" by (blast intro: le_anti_sym lemma_leq1 lemma_leq2) lemma (in sylow) sylow_thm: "∃H. subgroup H G & card(H) = p^a" apply (cut_tac lemma_A1, 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 prems) 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" apply (rule sylow.sylow_thm [of G p a m]) apply (simp add: sylow_eq sylow_axioms_def) done end
lemma RelM_refl:
refl calM RelM
lemma RelM_sym:
sym RelM
lemma RelM_trans:
trans RelM
lemma RelM_equiv:
equiv calM RelM
lemma M_subset_calM_prep:
M' ∈ calM // RelM ==> M' ⊆ calM
lemma M_subset_calM:
M ⊆ calM
lemma card_M1:
card M1 = p ^ a
lemma card_nonempty:
0 < card S ==> S ≠ {}
lemma exists_x_in_M1:
∃x. x ∈ M1
lemma M1_subset_G:
M1 ⊆ carrier G
lemma M1_inj_H:
∃f∈H -> M1. inj_on f H
lemma EmptyNotInEquivSet:
{} ∉ calM // RelM
lemma existsM1inM:
M ∈ calM // RelM ==> ∃M1. M1 ∈ M
lemma zero_less_o_G:
0 < Coset.order G
lemma zero_less_m:
0 < m
lemma card_calM:
card calM = p ^ a * m choose p ^ a
lemma zero_less_card_calM:
0 < card calM
lemma max_p_div_calM:
¬ p ^ Suc (exponent p m) dvd card calM
lemma finite_calM:
finite calM
lemma lemma_A1:
∃M∈calM // RelM. ¬ p ^ Suc (exponent p m) dvd card M
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
lemma H_not_empty:
H ≠ {}
lemma H_is_subgroup:
subgroup H G
lemma rcosetGM1g_subset_G:
[| g ∈ carrier G; x ∈ M1 #> g |] ==> x ∈ carrier G
lemma finite_M1:
finite M1
lemma finite_rcosetGM1g:
g ∈ carrier G ==> finite (M1 #> g)
lemma M1_cardeq_rcosetGM1g:
g ∈ carrier G ==> card (M1 #> g) = card M1
lemma M1_RelM_rcosetGM1g:
g ∈ carrier G ==> (M1, M1 #> g) ∈ RelM
lemma ElemClassEquiv:
[| equiv A r; C ∈ A // r |] ==> ∀x∈C. ∀y∈C. (x, y) ∈ r
lemma M_elem_map:
M2.0 ∈ M ==> ∃g. g ∈ carrier G ∧ M1 #> g = M2.0
lemma M_elem_map_carrier:
M2.2 ∈ M ==> (SOME x. x ∈ carrier G ∧ M1 #> x = M2.2) ∈ carrier G
lemma M_elem_map_eq:
M2.2 ∈ M ==> M1 #> (SOME x. x ∈ carrier G ∧ M1 #> x = M2.2) = M2.2
lemma M_funcset_rcosets_H:
(λx∈M. H #> (SOME g. g ∈ carrier G ∧ M1 #> g = x)) ∈ M -> rcosets H
lemma inj_M_GmodH:
∃f∈M -> rcosets H. inj_on f M
lemma H_elem_map:
H1.0 ∈ rcosets H ==> ∃g. g ∈ carrier G ∧ H #> g = H1.0
lemma H_elem_map_carrier:
H1.2 ∈ rcosets H ==> (SOME x. x ∈ carrier G ∧ H #> x = H1.2) ∈ carrier G
lemma H_elem_map_eq:
H1.2 ∈ rcosets H ==> H #> (SOME x. x ∈ carrier G ∧ H #> x = H1.2) = H1.2
lemma EquivElemClass:
[| equiv A r; M ∈ A // r; M1.0 ∈ M; (M1.0, M2.0) ∈ r |] ==> M2.0 ∈ M
lemma rcosets_H_funcset_M:
(λC∈rcosets H. M1 #> (SOME g. g ∈ carrier G ∧ H #> g = C)) ∈ rcosets H -> M
lemma inj_GmodH_M:
∃g∈rcosets H -> M. inj_on g (rcosets H)
lemma calM_subset_PowG:
calM ⊆ Pow (carrier G)
lemma finite_M:
finite M
lemma cardMeqIndexH:
card M = card (rcosets H)
lemma index_lem:
card M * card H = Coset.order G
lemma lemma_leq1:
p ^ a ≤ card H
lemma lemma_leq2:
card H ≤ p ^ a
lemma card_H_eq:
card H = p ^ a
lemma sylow_thm:
∃H. subgroup H G ∧ card H = p ^ a
lemma sylow_eq:
sylow G p a m = (group G ∧ sylow_axioms G p a m)
theorem sylow_thm:
[| prime p; group G; Coset.order G = p ^ a * m; finite (carrier G) |]
==> ∃H. subgroup H G ∧ card H = p ^ a