author | wenzelm |
Sun, 11 Sep 2016 00:13:25 +0200 | |
changeset 63832 | a400b127853c |
parent 63537 | 831816778409 |
child 64912 | 68f0465d956b |
permissions | -rw-r--r-- |
14706 | 1 |
(* Title: HOL/Algebra/Sylow.thy |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
2 |
Author: Florian Kammueller, with new proofs by L C Paulson |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
3 |
*) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
4 |
|
35849 | 5 |
theory Sylow |
6 |
imports Coset Exponent |
|
7 |
begin |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
8 |
|
61382 | 9 |
text \<open> |
58622 | 10 |
See also @{cite "Kammueller-Paulson:1999"}. |
61382 | 11 |
\<close> |
14706 | 12 |
|
61382 | 13 |
text\<open>The combinatorial argument is in theory Exponent\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
14 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
15 |
lemma le_extend_mult: |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
16 |
fixes c::nat shows "\<lbrakk>0 < c; a \<le> b\<rbrakk> \<Longrightarrow> a \<le> b * c" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
17 |
by (metis divisors_zero dvd_triv_left leI less_le_trans nat_dvd_not_less zero_less_iff_neq_zero) |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
18 |
|
14747 | 19 |
locale sylow = group + |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
20 |
fixes p and a and m and calM and RelM |
16663 | 21 |
assumes prime_p: "prime p" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
22 |
and order_G: "order(G) = (p^a) * m" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
23 |
and finite_G [iff]: "finite (carrier G)" |
14747 | 24 |
defines "calM == {s. s \<subseteq> carrier(G) & card(s) = p^a}" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
25 |
and "RelM == {(N1,N2). N1 \<in> calM & N2 \<in> calM & |
14666 | 26 |
(\<exists>g \<in> carrier(G). N1 = (N2 #> g) )}" |
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
27 |
begin |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
28 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
29 |
lemma RelM_refl_on: "refl_on calM RelM" |
30198 | 30 |
apply (auto simp add: refl_on_def RelM_def calM_def) |
14666 | 31 |
apply (blast intro!: coset_mult_one [symmetric]) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
32 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
33 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
34 |
lemma RelM_sym: "sym RelM" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
35 |
proof (unfold sym_def RelM_def, clarify) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
36 |
fix y g |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
37 |
assume "y \<in> calM" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
38 |
and g: "g \<in> carrier G" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
39 |
hence "y = y #> g #> (inv g)" by (simp add: coset_mult_assoc calM_def) |
41541 | 40 |
thus "\<exists>g'\<in>carrier G. y = y #> g #> g'" by (blast intro: g) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
41 |
qed |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
42 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
43 |
lemma RelM_trans: "trans RelM" |
14666 | 44 |
by (auto simp add: trans_def RelM_def calM_def coset_mult_assoc) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
45 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
46 |
lemma RelM_equiv: "equiv calM RelM" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
47 |
apply (unfold equiv_def) |
30198 | 48 |
apply (blast intro: RelM_refl_on RelM_sym RelM_trans) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
49 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
50 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
51 |
lemma M_subset_calM_prep: "M' \<in> calM // RelM ==> M' \<subseteq> calM" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
52 |
apply (unfold RelM_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
53 |
apply (blast elim!: quotientE) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
54 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
55 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
56 |
end |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
16663
diff
changeset
|
57 |
|
61382 | 58 |
subsection\<open>Main Part of the Proof\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
59 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
60 |
locale sylow_central = sylow + |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
61 |
fixes H and M1 and M |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
62 |
assumes M_in_quot: "M \<in> calM // RelM" |
63537
831816778409
Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents:
63534
diff
changeset
|
63 |
and not_dvd_M: "~(p ^ Suc(multiplicity p m) dvd card(M))" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
64 |
and M1_in_M: "M1 \<in> M" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
65 |
defines "H == {g. g\<in>carrier G & M1 #> g = M1}" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
66 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
67 |
begin |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
68 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
69 |
lemma M_subset_calM: "M \<subseteq> calM" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
70 |
by (rule M_in_quot [THEN M_subset_calM_prep]) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
71 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
72 |
lemma card_M1: "card(M1) = p^a" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
73 |
using M1_in_M M_subset_calM calM_def by blast |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
74 |
|
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
75 |
lemma exists_x_in_M1: "\<exists>x. x \<in> M1" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
76 |
using prime_p [THEN prime_gt_Suc_0_nat] card_M1 |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
77 |
by (metis Suc_lessD card_eq_0_iff empty_subsetI equalityI gr_implies_not0 nat_zero_less_power_iff subsetI) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
78 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
79 |
lemma M1_subset_G [simp]: "M1 \<subseteq> carrier G" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
80 |
using M1_in_M M_subset_calM calM_def mem_Collect_eq subsetCE by blast |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
81 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
82 |
lemma M1_inj_H: "\<exists>f \<in> H\<rightarrow>M1. inj_on f H" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
83 |
proof - |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
84 |
from exists_x_in_M1 obtain m1 where m1M: "m1 \<in> M1".. |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
85 |
have m1G: "m1 \<in> carrier G" by (simp add: m1M M1_subset_G [THEN subsetD]) |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
86 |
show ?thesis |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
87 |
proof |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
88 |
show "inj_on (\<lambda>z\<in>H. m1 \<otimes> z) H" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
89 |
by (simp add: inj_on_def l_cancel [of m1 x y, THEN iffD1] H_def m1G) |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
90 |
show "restrict (op \<otimes> m1) H \<in> H \<rightarrow> M1" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
91 |
proof (rule restrictI) |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
92 |
fix z assume zH: "z \<in> H" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
93 |
show "m1 \<otimes> z \<in> M1" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
94 |
proof - |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
95 |
from zH |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
96 |
have zG: "z \<in> carrier G" and M1zeq: "M1 #> z = M1" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
97 |
by (auto simp add: H_def) |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
98 |
show ?thesis |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
99 |
by (rule subst [OF M1zeq], simp add: m1M zG rcosI) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
100 |
qed |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
101 |
qed |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
102 |
qed |
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
103 |
qed |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
104 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
105 |
end |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
106 |
|
63167 | 107 |
subsection\<open>Discharging the Assumptions of \<open>sylow_central\<close>\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
108 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
109 |
context sylow |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
110 |
begin |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
111 |
|
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
112 |
lemma EmptyNotInEquivSet: "{} \<notin> calM // RelM" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
113 |
by (blast elim!: quotientE dest: RelM_equiv [THEN equiv_class_self]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
114 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
115 |
lemma existsM1inM: "M \<in> calM // RelM ==> \<exists>M1. M1 \<in> M" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
116 |
using RelM_equiv equiv_Eps_in by blast |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
117 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
118 |
lemma zero_less_o_G: "0 < order(G)" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
119 |
by (simp add: order_def card_gt_0_iff carrier_not_empty) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
120 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
121 |
lemma zero_less_m: "m > 0" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
122 |
using zero_less_o_G by (simp add: order_G) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
123 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
124 |
lemma card_calM: "card(calM) = (p^a) * m choose p^a" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
125 |
by (simp add: calM_def n_subsets order_G [symmetric] order_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
126 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
127 |
lemma zero_less_card_calM: "card calM > 0" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
128 |
by (simp add: card_calM zero_less_binomial le_extend_mult zero_less_m) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
129 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
130 |
lemma max_p_div_calM: |
63537
831816778409
Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents:
63534
diff
changeset
|
131 |
"~ (p ^ Suc(multiplicity p m) dvd card(calM))" |
63534
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
132 |
proof |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
133 |
assume "p ^ Suc (multiplicity p m) dvd card calM" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
134 |
with zero_less_card_calM prime_p |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
135 |
have "Suc (multiplicity p m) \<le> multiplicity p (card calM)" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
136 |
by (intro multiplicity_geI) auto |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
137 |
hence "multiplicity p m < multiplicity p (card calM)" by simp |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
138 |
also have "multiplicity p m = multiplicity p (card calM)" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
139 |
by (simp add: const_p_fac prime_p zero_less_m card_calM) |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63167
diff
changeset
|
140 |
finally show False by simp |
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
141 |
qed |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
142 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
143 |
lemma finite_calM: "finite calM" |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
144 |
unfolding calM_def |
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
145 |
by (rule_tac B = "Pow (carrier G) " in finite_subset) auto |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
146 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
147 |
lemma lemma_A1: |
63537
831816778409
Removed redundant material related to primes
eberlm <eberlm@in.tum.de>
parents:
63534
diff
changeset
|
148 |
"\<exists>M \<in> calM // RelM. ~ (p ^ Suc(multiplicity p m) dvd card(M))" |
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
149 |
using RelM_equiv equiv_imp_dvd_card finite_calM max_p_div_calM by blast |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
150 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
151 |
end |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
152 |
|
61382 | 153 |
subsubsection\<open>Introduction and Destruct Rules for @{term H}\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
154 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
155 |
lemma (in sylow_central) H_I: "[|g \<in> carrier G; M1 #> g = M1|] ==> g \<in> H" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
156 |
by (simp add: H_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
157 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
158 |
lemma (in sylow_central) H_into_carrier_G: "x \<in> H ==> x \<in> carrier G" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
159 |
by (simp add: H_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
160 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
161 |
lemma (in sylow_central) in_H_imp_eq: "g : H ==> M1 #> g = M1" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
162 |
by (simp add: H_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
163 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
164 |
lemma (in sylow_central) H_m_closed: "[| x\<in>H; y\<in>H|] ==> x \<otimes> y \<in> H" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
165 |
apply (unfold H_def) |
41541 | 166 |
apply (simp add: coset_mult_assoc [symmetric]) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
167 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
168 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
169 |
lemma (in sylow_central) H_not_empty: "H \<noteq> {}" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
170 |
apply (simp add: H_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
171 |
apply (rule exI [of _ \<one>], simp) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
172 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
173 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
174 |
lemma (in sylow_central) H_is_subgroup: "subgroup H G" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
175 |
apply (rule subgroupI) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
176 |
apply (rule subsetI) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
177 |
apply (erule H_into_carrier_G) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
178 |
apply (rule H_not_empty) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
179 |
apply (simp add: H_def, clarify) |
59807 | 180 |
apply (erule_tac P = "%z. lhs(z) = M1" for lhs in subst) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
181 |
apply (simp add: coset_mult_assoc ) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
182 |
apply (blast intro: H_m_closed) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
183 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
184 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
185 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
186 |
lemma (in sylow_central) rcosetGM1g_subset_G: |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
187 |
"[| g \<in> carrier G; x \<in> M1 #> g |] ==> x \<in> carrier G" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
188 |
by (blast intro: M1_subset_G [THEN r_coset_subset_G, THEN subsetD]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
189 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
190 |
lemma (in sylow_central) finite_M1: "finite M1" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
191 |
by (rule finite_subset [OF M1_subset_G finite_G]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
192 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
193 |
lemma (in sylow_central) finite_rcosetGM1g: "g\<in>carrier G ==> finite (M1 #> g)" |
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
194 |
using rcosetGM1g_subset_G finite_G M1_subset_G cosets_finite rcosetsI by blast |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
195 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
196 |
lemma (in sylow_central) M1_cardeq_rcosetGM1g: |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
197 |
"g \<in> carrier G ==> card(M1 #> g) = card(M1)" |
41541 | 198 |
by (simp (no_asm_simp) add: card_cosets_equal rcosetsI) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
199 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
200 |
lemma (in sylow_central) M1_RelM_rcosetGM1g: |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
201 |
"g \<in> carrier G ==> (M1, M1 #> g) \<in> RelM" |
55157 | 202 |
apply (simp add: RelM_def calM_def card_M1) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
203 |
apply (rule conjI) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
204 |
apply (blast intro: rcosetGM1g_subset_G) |
55157 | 205 |
apply (simp add: card_M1 M1_cardeq_rcosetGM1g) |
206 |
apply (metis M1_subset_G coset_mult_assoc coset_mult_one r_inv_ex) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
207 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
208 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
209 |
|
61382 | 210 |
subsection\<open>Equal Cardinalities of @{term M} and the Set of Cosets\<close> |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
211 |
|
61382 | 212 |
text\<open>Injections between @{term M} and @{term "rcosets\<^bsub>G\<^esub> H"} show that |
213 |
their cardinalities are equal.\<close> |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
214 |
|
14666 | 215 |
lemma ElemClassEquiv: |
14963 | 216 |
"[| equiv A r; C \<in> A // r |] ==> \<forall>x \<in> C. \<forall>y \<in> C. (x,y)\<in>r" |
217 |
by (unfold equiv_def quotient_def sym_def trans_def, blast) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
218 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
219 |
lemma (in sylow_central) M_elem_map: |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
220 |
"M2 \<in> M ==> \<exists>g. g \<in> carrier G & M1 #> g = M2" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
221 |
apply (cut_tac M1_in_M M_in_quot [THEN RelM_equiv [THEN ElemClassEquiv]]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
222 |
apply (simp add: RelM_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
223 |
apply (blast dest!: bspec) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
224 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
225 |
|
14666 | 226 |
lemmas (in sylow_central) M_elem_map_carrier = |
227 |
M_elem_map [THEN someI_ex, THEN conjunct1] |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
228 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
229 |
lemmas (in sylow_central) M_elem_map_eq = |
14666 | 230 |
M_elem_map [THEN someI_ex, THEN conjunct2] |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
231 |
|
14963 | 232 |
lemma (in sylow_central) M_funcset_rcosets_H: |
233 |
"(%x:M. H #> (SOME g. g \<in> carrier G & M1 #> g = x)) \<in> M \<rightarrow> rcosets H" |
|
55157 | 234 |
by (metis (lifting) H_is_subgroup M_elem_map_carrier rcosetsI restrictI subgroup_imp_subset) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
235 |
|
62410
2fc7a8d9c529
partial tidy-up of Sylow's theorem
paulson <lp15@cam.ac.uk>
parents:
61382
diff
changeset
|
236 |
lemma (in sylow_central) inj_M_GmodH: "\<exists>f \<in> M \<rightarrow> rcosets H. inj_on f M" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
237 |
apply (rule bexI) |
14963 | 238 |
apply (rule_tac [2] M_funcset_rcosets_H) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
239 |
apply (rule inj_onI, simp) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
240 |
apply (rule trans [OF _ M_elem_map_eq]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
241 |
prefer 2 apply assumption |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
242 |
apply (rule M_elem_map_eq [symmetric, THEN trans], assumption) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
243 |
apply (rule coset_mult_inv1) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
244 |
apply (erule_tac [2] M_elem_map_carrier)+ |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
245 |
apply (rule_tac [2] M1_subset_G) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
246 |
apply (rule coset_join1 [THEN in_H_imp_eq]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
247 |
apply (rule_tac [3] H_is_subgroup) |
41541 | 248 |
prefer 2 apply (blast intro: M_elem_map_carrier) |
26806 | 249 |
apply (simp add: coset_mult_inv2 H_def M_elem_map_carrier subset_eq) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
250 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
251 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
252 |
|
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subsubsection\<open>The Opposite Injection\<close> |
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|
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lemma (in sylow_central) H_elem_map: |
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"H1 \<in> rcosets H ==> \<exists>g. g \<in> carrier G & H #> g = H1" |
257 |
by (auto simp add: RCOSETS_def) |
|
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|
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lemmas (in sylow_central) H_elem_map_carrier = |
260 |
H_elem_map [THEN someI_ex, THEN conjunct1] |
|
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|
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lemmas (in sylow_central) H_elem_map_eq = |
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H_elem_map [THEN someI_ex, THEN conjunct2] |
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264 |
|
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lemma (in sylow_central) rcosets_H_funcset_M: |
266 |
"(\<lambda>C \<in> rcosets H. M1 #> (@g. g \<in> carrier G \<and> H #> g = C)) \<in> rcosets H \<rightarrow> M" |
|
267 |
apply (simp add: RCOSETS_def) |
|
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apply (fast intro: someI2 |
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intro!: M1_in_M in_quotient_imp_closed [OF RelM_equiv M_in_quot _ M1_RelM_rcosetGM1g]) |
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done |
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|
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text\<open>close to a duplicate of \<open>inj_M_GmodH\<close>\<close> |
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lemma (in sylow_central) inj_GmodH_M: |
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"\<exists>g \<in> rcosets H\<rightarrow>M. inj_on g (rcosets H)" |
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|
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apply (rule bexI) |
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apply (rule_tac [2] rcosets_H_funcset_M) |
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|
277 |
apply (rule inj_onI) |
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|
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apply (simp) |
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|
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apply (rule trans [OF _ H_elem_map_eq]) |
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prefer 2 apply assumption |
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parents:
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apply (rule H_elem_map_eq [symmetric, THEN trans], assumption) |
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parents:
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|
282 |
apply (rule coset_mult_inv1) |
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parents:
diff
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|
283 |
apply (erule_tac [2] H_elem_map_carrier)+ |
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parents:
diff
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|
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apply (rule_tac [2] H_is_subgroup [THEN subgroup.subset]) |
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parents:
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|
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apply (rule coset_join2) |
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apply (blast intro: H_elem_map_carrier) |
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apply (rule H_is_subgroup) |
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apply (simp add: H_I coset_mult_inv2 H_elem_map_carrier) |
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|
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done |
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|
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lemma (in sylow_central) calM_subset_PowG: "calM \<subseteq> Pow(carrier G)" |
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parents:
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|
292 |
by (auto simp add: calM_def) |
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parents:
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changeset
|
293 |
|
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changeset
|
294 |
|
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|
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lemma (in sylow_central) finite_M: "finite M" |
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by (metis M_subset_calM finite_calM rev_finite_subset) |
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|
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|
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lemma (in sylow_central) cardMeqIndexH: "card(M) = card(rcosets H)" |
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apply (insert inj_M_GmodH inj_GmodH_M) |
300 |
apply (blast intro: card_bij finite_M H_is_subgroup |
|
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rcosets_subset_PowG [THEN finite_subset] |
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finite_Pow_iff [THEN iffD2]) |
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|
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done |
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changeset
|
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|
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|
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lemma (in sylow_central) index_lem: "card(M) * card(H) = order(G)" |
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parents:
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changeset
|
306 |
by (simp add: cardMeqIndexH lagrange H_is_subgroup) |
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|
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|
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lemma (in sylow_central) lemma_leq1: "p^a \<le> card(H)" |
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|
309 |
apply (rule dvd_imp_le) |
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eberlm <eberlm@in.tum.de>
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changeset
|
310 |
apply (rule div_combine [OF prime_imp_prime_elem[OF prime_p] not_dvd_M]) |
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|
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prefer 2 apply (blast intro: subgroup.finite_imp_card_positive H_is_subgroup) |
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Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
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changeset
|
312 |
apply (simp add: index_lem order_G power_add mult_dvd_mono multiplicity_dvd |
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|
313 |
zero_less_m) |
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changeset
|
314 |
done |
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|
315 |
|
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lemma (in sylow_central) lemma_leq2: "card(H) \<le> p^a" |
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changeset
|
317 |
apply (subst card_M1 [symmetric]) |
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|
318 |
apply (cut_tac M1_inj_H) |
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apply (blast intro!: M1_subset_G intro: |
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|
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card_inj H_into_carrier_G finite_subset [OF _ finite_G]) |
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parents:
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changeset
|
321 |
done |
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changeset
|
322 |
|
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|
323 |
lemma (in sylow_central) card_H_eq: "card(H) = p^a" |
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by (blast intro: le_antisym lemma_leq1 lemma_leq2) |
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changeset
|
325 |
|
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|
326 |
lemma (in sylow) sylow_thm: "\<exists>H. subgroup H G & card(H) = p^a" |
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apply (cut_tac lemma_A1, clarify) |
328 |
apply (frule existsM1inM, clarify) |
|
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|
329 |
apply (subgoal_tac "sylow_central G p a m M1 M") |
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parents:
diff
changeset
|
330 |
apply (blast dest: sylow_central.H_is_subgroup sylow_central.card_H_eq) |
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apply (simp add: sylow_central_def sylow_central_axioms_def sylow_axioms calM_def RelM_def) |
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|
332 |
done |
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changeset
|
333 |
|
61382 | 334 |
text\<open>Needed because the locale's automatic definition refers to |
14666 | 335 |
@{term "semigroup G"} and @{term "group_axioms G"} rather than |
61382 | 336 |
simply to @{term "group G"}.\<close> |
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|
337 |
lemma sylow_eq: "sylow G p a m = (group G & sylow_axioms G p a m)" |
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changeset
|
338 |
by (simp add: sylow_def group_def) |
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diff
changeset
|
339 |
|
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ballarin
parents:
16663
diff
changeset
|
340 |
|
61382 | 341 |
subsection \<open>Sylow's Theorem\<close> |
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ballarin
parents:
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changeset
|
342 |
|
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|
343 |
theorem sylow_thm: |
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"[| prime p; group(G); order(G) = (p^a) * m; finite (carrier G)|] |
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|
345 |
==> \<exists>H. subgroup H G & card(H) = p^a" |
cf947d1ec5ff
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paulson
parents:
diff
changeset
|
346 |
apply (rule sylow.sylow_thm [of G p a m]) |
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apply (simp add: sylow_eq sylow_axioms_def) |
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changeset
|
348 |
done |
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parents:
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changeset
|
349 |
|
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|
350 |
end |