src/HOL/Algebra/Sylow.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 16417 9bc16273c2d4 child 16663 13e9c402308b permissions -rw-r--r--
Constant "If" is now local
```     1 (*  Title:      HOL/Algebra/Sylow.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Florian Kammueller, with new proofs by L C Paulson
```
```     4 *)
```
```     5
```
```     6 header {* Sylow's theorem *}
```
```     7
```
```     8 theory Sylow imports Coset begin
```
```     9
```
```    10 text {*
```
```    11   See also \cite{Kammueller-Paulson:1999}.
```
```    12 *}
```
```    13
```
```    14 text{*The combinatorial argument is in theory Exponent*}
```
```    15
```
```    16 locale sylow = group +
```
```    17   fixes p and a and m and calM and RelM
```
```    18   assumes prime_p:   "p \<in> prime"
```
```    19       and order_G:   "order(G) = (p^a) * m"
```
```    20       and finite_G [iff]:  "finite (carrier G)"
```
```    21   defines "calM == {s. s \<subseteq> carrier(G) & card(s) = p^a}"
```
```    22       and "RelM == {(N1,N2). N1 \<in> calM & N2 \<in> calM &
```
```    23                              (\<exists>g \<in> carrier(G). N1 = (N2 #> g) )}"
```
```    24
```
```    25 lemma (in sylow) RelM_refl: "refl calM RelM"
```
```    26 apply (auto simp add: refl_def RelM_def calM_def)
```
```    27 apply (blast intro!: coset_mult_one [symmetric])
```
```    28 done
```
```    29
```
```    30 lemma (in sylow) RelM_sym: "sym RelM"
```
```    31 proof (unfold sym_def RelM_def, clarify)
```
```    32   fix y g
```
```    33   assume   "y \<in> calM"
```
```    34     and g: "g \<in> carrier G"
```
```    35   hence "y = y #> g #> (inv g)" by (simp add: coset_mult_assoc calM_def)
```
```    36   thus "\<exists>g'\<in>carrier G. y = y #> g #> g'"
```
```    37    by (blast intro: g inv_closed)
```
```    38 qed
```
```    39
```
```    40 lemma (in sylow) RelM_trans: "trans RelM"
```
```    41 by (auto simp add: trans_def RelM_def calM_def coset_mult_assoc)
```
```    42
```
```    43 lemma (in sylow) RelM_equiv: "equiv calM RelM"
```
```    44 apply (unfold equiv_def)
```
```    45 apply (blast intro: RelM_refl RelM_sym RelM_trans)
```
```    46 done
```
```    47
```
```    48 lemma (in sylow) M_subset_calM_prep: "M' \<in> calM // RelM  ==> M' \<subseteq> calM"
```
```    49 apply (unfold RelM_def)
```
```    50 apply (blast elim!: quotientE)
```
```    51 done
```
```    52
```
```    53 subsection{*Main Part of the Proof*}
```
```    54
```
```    55
```
```    56 locale sylow_central = sylow +
```
```    57   fixes H and M1 and M
```
```    58   assumes M_in_quot:  "M \<in> calM // RelM"
```
```    59       and not_dvd_M:  "~(p ^ Suc(exponent p m) dvd card(M))"
```
```    60       and M1_in_M:    "M1 \<in> M"
```
```    61   defines "H == {g. g\<in>carrier G & M1 #> g = M1}"
```
```    62
```
```    63 lemma (in sylow_central) M_subset_calM: "M \<subseteq> calM"
```
```    64 by (rule M_in_quot [THEN M_subset_calM_prep])
```
```    65
```
```    66 lemma (in sylow_central) card_M1: "card(M1) = p^a"
```
```    67 apply (cut_tac M_subset_calM M1_in_M)
```
```    68 apply (simp add: calM_def, blast)
```
```    69 done
```
```    70
```
```    71 lemma card_nonempty: "0 < card(S) ==> S \<noteq> {}"
```
```    72 by force
```
```    73
```
```    74 lemma (in sylow_central) exists_x_in_M1: "\<exists>x. x\<in>M1"
```
```    75 apply (subgoal_tac "0 < card M1")
```
```    76  apply (blast dest: card_nonempty)
```
```    77 apply (cut_tac prime_p [THEN prime_imp_one_less])
```
```    78 apply (simp (no_asm_simp) add: card_M1)
```
```    79 done
```
```    80
```
```    81 lemma (in sylow_central) M1_subset_G [simp]: "M1 \<subseteq> carrier G"
```
```    82 apply (rule subsetD [THEN PowD])
```
```    83 apply (rule_tac [2] M1_in_M)
```
```    84 apply (rule M_subset_calM [THEN subset_trans])
```
```    85 apply (auto simp add: calM_def)
```
```    86 done
```
```    87
```
```    88 lemma (in sylow_central) M1_inj_H: "\<exists>f \<in> H\<rightarrow>M1. inj_on f H"
```
```    89   proof -
```
```    90     from exists_x_in_M1 obtain m1 where m1M: "m1 \<in> M1"..
```
```    91     have m1G: "m1 \<in> carrier G" by (simp add: m1M M1_subset_G [THEN subsetD])
```
```    92     show ?thesis
```
```    93     proof
```
```    94       show "inj_on (\<lambda>z\<in>H. m1 \<otimes> z) H"
```
```    95         by (simp add: inj_on_def l_cancel [of m1 x y, THEN iffD1] H_def m1G)
```
```    96       show "restrict (op \<otimes> m1) H \<in> H \<rightarrow> M1"
```
```    97       proof (rule restrictI)
```
```    98         fix z assume zH: "z \<in> H"
```
```    99         show "m1 \<otimes> z \<in> M1"
```
```   100         proof -
```
```   101           from zH
```
```   102           have zG: "z \<in> carrier G" and M1zeq: "M1 #> z = M1"
```
```   103             by (auto simp add: H_def)
```
```   104           show ?thesis
```
```   105             by (rule subst [OF M1zeq], simp add: m1M zG rcosI)
```
```   106         qed
```
```   107       qed
```
```   108     qed
```
```   109   qed
```
```   110
```
```   111
```
```   112 subsection{*Discharging the Assumptions of @{text sylow_central}*}
```
```   113
```
```   114 lemma (in sylow) EmptyNotInEquivSet: "{} \<notin> calM // RelM"
```
```   115 by (blast elim!: quotientE dest: RelM_equiv [THEN equiv_class_self])
```
```   116
```
```   117 lemma (in sylow) existsM1inM: "M \<in> calM // RelM ==> \<exists>M1. M1 \<in> M"
```
```   118 apply (subgoal_tac "M \<noteq> {}")
```
```   119  apply blast
```
```   120 apply (cut_tac EmptyNotInEquivSet, blast)
```
```   121 done
```
```   122
```
```   123 lemma (in sylow) zero_less_o_G: "0 < order(G)"
```
```   124 apply (unfold order_def)
```
```   125 apply (blast intro: one_closed zero_less_card_empty)
```
```   126 done
```
```   127
```
```   128 lemma (in sylow) zero_less_m: "0 < m"
```
```   129 apply (cut_tac zero_less_o_G)
```
```   130 apply (simp add: order_G)
```
```   131 done
```
```   132
```
```   133 lemma (in sylow) card_calM: "card(calM) = (p^a) * m choose p^a"
```
```   134 by (simp add: calM_def n_subsets order_G [symmetric] order_def)
```
```   135
```
```   136 lemma (in sylow) zero_less_card_calM: "0 < card calM"
```
```   137 by (simp add: card_calM zero_less_binomial le_extend_mult zero_less_m)
```
```   138
```
```   139 lemma (in sylow) max_p_div_calM:
```
```   140      "~ (p ^ Suc(exponent p m) dvd card(calM))"
```
```   141 apply (subgoal_tac "exponent p m = exponent p (card calM) ")
```
```   142  apply (cut_tac zero_less_card_calM prime_p)
```
```   143  apply (force dest: power_Suc_exponent_Not_dvd)
```
```   144 apply (simp add: card_calM zero_less_m [THEN const_p_fac])
```
```   145 done
```
```   146
```
```   147 lemma (in sylow) finite_calM: "finite calM"
```
```   148 apply (unfold calM_def)
```
```   149 apply (rule_tac B = "Pow (carrier G) " in finite_subset)
```
```   150 apply auto
```
```   151 done
```
```   152
```
```   153 lemma (in sylow) lemma_A1:
```
```   154      "\<exists>M \<in> calM // RelM. ~ (p ^ Suc(exponent p m) dvd card(M))"
```
```   155 apply (rule max_p_div_calM [THEN contrapos_np])
```
```   156 apply (simp add: finite_calM equiv_imp_dvd_card [OF _ RelM_equiv])
```
```   157 done
```
```   158
```
```   159
```
```   160 subsubsection{*Introduction and Destruct Rules for @{term H}*}
```
```   161
```
```   162 lemma (in sylow_central) H_I: "[|g \<in> carrier G; M1 #> g = M1|] ==> g \<in> H"
```
```   163 by (simp add: H_def)
```
```   164
```
```   165 lemma (in sylow_central) H_into_carrier_G: "x \<in> H ==> x \<in> carrier G"
```
```   166 by (simp add: H_def)
```
```   167
```
```   168 lemma (in sylow_central) in_H_imp_eq: "g : H ==> M1 #> g = M1"
```
```   169 by (simp add: H_def)
```
```   170
```
```   171 lemma (in sylow_central) H_m_closed: "[| x\<in>H; y\<in>H|] ==> x \<otimes> y \<in> H"
```
```   172 apply (unfold H_def)
```
```   173 apply (simp add: coset_mult_assoc [symmetric] m_closed)
```
```   174 done
```
```   175
```
```   176 lemma (in sylow_central) H_not_empty: "H \<noteq> {}"
```
```   177 apply (simp add: H_def)
```
```   178 apply (rule exI [of _ \<one>], simp)
```
```   179 done
```
```   180
```
```   181 lemma (in sylow_central) H_is_subgroup: "subgroup H G"
```
```   182 apply (rule subgroupI)
```
```   183 apply (rule subsetI)
```
```   184 apply (erule H_into_carrier_G)
```
```   185 apply (rule H_not_empty)
```
```   186 apply (simp add: H_def, clarify)
```
```   187 apply (erule_tac P = "%z. ?lhs(z) = M1" in subst)
```
```   188 apply (simp add: coset_mult_assoc )
```
```   189 apply (blast intro: H_m_closed)
```
```   190 done
```
```   191
```
```   192
```
```   193 lemma (in sylow_central) rcosetGM1g_subset_G:
```
```   194      "[| g \<in> carrier G; x \<in> M1 #>  g |] ==> x \<in> carrier G"
```
```   195 by (blast intro: M1_subset_G [THEN r_coset_subset_G, THEN subsetD])
```
```   196
```
```   197 lemma (in sylow_central) finite_M1: "finite M1"
```
```   198 by (rule finite_subset [OF M1_subset_G finite_G])
```
```   199
```
```   200 lemma (in sylow_central) finite_rcosetGM1g: "g\<in>carrier G ==> finite (M1 #> g)"
```
```   201 apply (rule finite_subset)
```
```   202 apply (rule subsetI)
```
```   203 apply (erule rcosetGM1g_subset_G, assumption)
```
```   204 apply (rule finite_G)
```
```   205 done
```
```   206
```
```   207 lemma (in sylow_central) M1_cardeq_rcosetGM1g:
```
```   208      "g \<in> carrier G ==> card(M1 #> g) = card(M1)"
```
```   209 by (simp (no_asm_simp) add: M1_subset_G card_cosets_equal rcosetsI)
```
```   210
```
```   211 lemma (in sylow_central) M1_RelM_rcosetGM1g:
```
```   212      "g \<in> carrier G ==> (M1, M1 #> g) \<in> RelM"
```
```   213 apply (simp (no_asm) add: RelM_def calM_def card_M1 M1_subset_G)
```
```   214 apply (rule conjI)
```
```   215  apply (blast intro: rcosetGM1g_subset_G)
```
```   216 apply (simp (no_asm_simp) add: card_M1 M1_cardeq_rcosetGM1g)
```
```   217 apply (rule bexI [of _ "inv g"])
```
```   218 apply (simp_all add: coset_mult_assoc M1_subset_G)
```
```   219 done
```
```   220
```
```   221
```
```   222 subsection{*Equal Cardinalities of @{term M} and the Set of Cosets*}
```
```   223
```
```   224 text{*Injections between @{term M} and @{term "rcosets\<^bsub>G\<^esub> H"} show that
```
```   225  their cardinalities are equal.*}
```
```   226
```
```   227 lemma ElemClassEquiv:
```
```   228      "[| equiv A r; C \<in> A // r |] ==> \<forall>x \<in> C. \<forall>y \<in> C. (x,y)\<in>r"
```
```   229 by (unfold equiv_def quotient_def sym_def trans_def, blast)
```
```   230
```
```   231 lemma (in sylow_central) M_elem_map:
```
```   232      "M2 \<in> M ==> \<exists>g. g \<in> carrier G & M1 #> g = M2"
```
```   233 apply (cut_tac M1_in_M M_in_quot [THEN RelM_equiv [THEN ElemClassEquiv]])
```
```   234 apply (simp add: RelM_def)
```
```   235 apply (blast dest!: bspec)
```
```   236 done
```
```   237
```
```   238 lemmas (in sylow_central) M_elem_map_carrier =
```
```   239         M_elem_map [THEN someI_ex, THEN conjunct1]
```
```   240
```
```   241 lemmas (in sylow_central) M_elem_map_eq =
```
```   242         M_elem_map [THEN someI_ex, THEN conjunct2]
```
```   243
```
```   244 lemma (in sylow_central) M_funcset_rcosets_H:
```
```   245      "(%x:M. H #> (SOME g. g \<in> carrier G & M1 #> g = x)) \<in> M \<rightarrow> rcosets H"
```
```   246 apply (rule rcosetsI [THEN restrictI])
```
```   247 apply (rule H_is_subgroup [THEN subgroup.subset])
```
```   248 apply (erule M_elem_map_carrier)
```
```   249 done
```
```   250
```
```   251 lemma (in sylow_central) inj_M_GmodH: "\<exists>f \<in> M\<rightarrow>rcosets H. inj_on f M"
```
```   252 apply (rule bexI)
```
```   253 apply (rule_tac [2] M_funcset_rcosets_H)
```
```   254 apply (rule inj_onI, simp)
```
```   255 apply (rule trans [OF _ M_elem_map_eq])
```
```   256 prefer 2 apply assumption
```
```   257 apply (rule M_elem_map_eq [symmetric, THEN trans], assumption)
```
```   258 apply (rule coset_mult_inv1)
```
```   259 apply (erule_tac [2] M_elem_map_carrier)+
```
```   260 apply (rule_tac [2] M1_subset_G)
```
```   261 apply (rule coset_join1 [THEN in_H_imp_eq])
```
```   262 apply (rule_tac [3] H_is_subgroup)
```
```   263 prefer 2 apply (blast intro: m_closed M_elem_map_carrier inv_closed)
```
```   264 apply (simp add: coset_mult_inv2 H_def M_elem_map_carrier subset_def)
```
```   265 done
```
```   266
```
```   267
```
```   268 subsubsection{*The opposite injection*}
```
```   269
```
```   270 lemma (in sylow_central) H_elem_map:
```
```   271      "H1 \<in> rcosets H ==> \<exists>g. g \<in> carrier G & H #> g = H1"
```
```   272 by (auto simp add: RCOSETS_def)
```
```   273
```
```   274 lemmas (in sylow_central) H_elem_map_carrier =
```
```   275         H_elem_map [THEN someI_ex, THEN conjunct1]
```
```   276
```
```   277 lemmas (in sylow_central) H_elem_map_eq =
```
```   278         H_elem_map [THEN someI_ex, THEN conjunct2]
```
```   279
```
```   280
```
```   281 lemma EquivElemClass:
```
```   282      "[|equiv A r; M \<in> A//r; M1\<in>M; (M1,M2) \<in> r |] ==> M2 \<in> M"
```
```   283 by (unfold equiv_def quotient_def sym_def trans_def, blast)
```
```   284
```
```   285
```
```   286 lemma (in sylow_central) rcosets_H_funcset_M:
```
```   287   "(\<lambda>C \<in> rcosets H. M1 #> (@g. g \<in> carrier G \<and> H #> g = C)) \<in> rcosets H \<rightarrow> M"
```
```   288 apply (simp add: RCOSETS_def)
```
```   289 apply (fast intro: someI2
```
```   290             intro!: restrictI M1_in_M
```
```   291               EquivElemClass [OF RelM_equiv M_in_quot _  M1_RelM_rcosetGM1g])
```
```   292 done
```
```   293
```
```   294 text{*close to a duplicate of @{text inj_M_GmodH}*}
```
```   295 lemma (in sylow_central) inj_GmodH_M:
```
```   296      "\<exists>g \<in> rcosets H\<rightarrow>M. inj_on g (rcosets H)"
```
```   297 apply (rule bexI)
```
```   298 apply (rule_tac [2] rcosets_H_funcset_M)
```
```   299 apply (rule inj_onI)
```
```   300 apply (simp)
```
```   301 apply (rule trans [OF _ H_elem_map_eq])
```
```   302 prefer 2 apply assumption
```
```   303 apply (rule H_elem_map_eq [symmetric, THEN trans], assumption)
```
```   304 apply (rule coset_mult_inv1)
```
```   305 apply (erule_tac [2] H_elem_map_carrier)+
```
```   306 apply (rule_tac [2] H_is_subgroup [THEN subgroup.subset])
```
```   307 apply (rule coset_join2)
```
```   308 apply (blast intro: m_closed inv_closed H_elem_map_carrier)
```
```   309 apply (rule H_is_subgroup)
```
```   310 apply (simp add: H_I coset_mult_inv2 M1_subset_G H_elem_map_carrier)
```
```   311 done
```
```   312
```
```   313 lemma (in sylow_central) calM_subset_PowG: "calM \<subseteq> Pow(carrier G)"
```
```   314 by (auto simp add: calM_def)
```
```   315
```
```   316
```
```   317 lemma (in sylow_central) finite_M: "finite M"
```
```   318 apply (rule finite_subset)
```
```   319 apply (rule M_subset_calM [THEN subset_trans])
```
```   320 apply (rule calM_subset_PowG, blast)
```
```   321 done
```
```   322
```
```   323 lemma (in sylow_central) cardMeqIndexH: "card(M) = card(rcosets H)"
```
```   324 apply (insert inj_M_GmodH inj_GmodH_M)
```
```   325 apply (blast intro: card_bij finite_M H_is_subgroup
```
```   326              rcosets_subset_PowG [THEN finite_subset]
```
```   327              finite_Pow_iff [THEN iffD2])
```
```   328 done
```
```   329
```
```   330 lemma (in sylow_central) index_lem: "card(M) * card(H) = order(G)"
```
```   331 by (simp add: cardMeqIndexH lagrange H_is_subgroup)
```
```   332
```
```   333 lemma (in sylow_central) lemma_leq1: "p^a \<le> card(H)"
```
```   334 apply (rule dvd_imp_le)
```
```   335  apply (rule div_combine [OF prime_p not_dvd_M])
```
```   336  prefer 2 apply (blast intro: subgroup.finite_imp_card_positive H_is_subgroup)
```
```   337 apply (simp add: index_lem order_G power_add mult_dvd_mono power_exponent_dvd
```
```   338                  zero_less_m)
```
```   339 done
```
```   340
```
```   341 lemma (in sylow_central) lemma_leq2: "card(H) \<le> p^a"
```
```   342 apply (subst card_M1 [symmetric])
```
```   343 apply (cut_tac M1_inj_H)
```
```   344 apply (blast intro!: M1_subset_G intro:
```
```   345              card_inj H_into_carrier_G finite_subset [OF _ finite_G])
```
```   346 done
```
```   347
```
```   348 lemma (in sylow_central) card_H_eq: "card(H) = p^a"
```
```   349 by (blast intro: le_anti_sym lemma_leq1 lemma_leq2)
```
```   350
```
```   351 lemma (in sylow) sylow_thm: "\<exists>H. subgroup H G & card(H) = p^a"
```
```   352 apply (cut_tac lemma_A1, clarify)
```
```   353 apply (frule existsM1inM, clarify)
```
```   354 apply (subgoal_tac "sylow_central G p a m M1 M")
```
```   355  apply (blast dest:  sylow_central.H_is_subgroup sylow_central.card_H_eq)
```
```   356 apply (simp add: sylow_central_def sylow_central_axioms_def prems)
```
```   357 done
```
```   358
```
```   359 text{*Needed because the locale's automatic definition refers to
```
```   360    @{term "semigroup G"} and @{term "group_axioms G"} rather than
```
```   361   simply to @{term "group G"}.*}
```
```   362 lemma sylow_eq: "sylow G p a m = (group G & sylow_axioms G p a m)"
```
```   363 by (simp add: sylow_def group_def)
```
```   364
```
```   365 theorem sylow_thm:
```
```   366      "[|p \<in> prime;  group(G);  order(G) = (p^a) * m; finite (carrier G)|]
```
```   367       ==> \<exists>H. subgroup H G & card(H) = p^a"
```
```   368 apply (rule sylow.sylow_thm [of G p a m])
```
```   369 apply (simp add: sylow_eq sylow_axioms_def)
```
```   370 done
```
```   371
```
```   372 end
```
```   373
```