author | paulson |
Fri, 14 May 2004 16:50:33 +0200 | |
changeset 14747 | 2eaff69d3d10 |
parent 14706 | 71590b7733b7 |
child 14761 | 28b5eb4a867f |
permissions | -rw-r--r-- |
14706 | 1 |
(* Title: HOL/Algebra/Coset.thy |
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ID: $Id$ |
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Author: Florian Kammueller, with new proofs by L C Paulson |
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*) |
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|
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header{*Cosets and Quotient Groups*} |
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|
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theory Coset = Group + Exponent: |
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declare (in group) l_inv [simp] and r_inv [simp] |
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constdefs (structure G) |
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r_coset :: "[_, 'a set, 'a] => 'a set" (infixl "#>\<index>" 60) |
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"H #> a == (% x. x \<otimes> a) ` H" |
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l_coset :: "[_, 'a, 'a set] => 'a set" (infixl "<#\<index>" 60) |
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"a <# H == (% x. a \<otimes> x) ` H" |
|
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rcosets :: "[_, 'a set] => ('a set)set" |
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"rcosets G H == r_coset G H ` (carrier G)" |
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set_mult :: "[_, 'a set ,'a set] => 'a set" (infixl "<#>\<index>" 60) |
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"H <#> K == (%(x,y). x \<otimes> y) ` (H \<times> K)" |
|
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set_inv :: "[_,'a set] => 'a set" |
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"set_inv G H == m_inv G ` H" |
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normal :: "['a set, _] => bool" (infixl "<|" 60) |
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"normal H G == subgroup H G & |
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(\<forall>x \<in> carrier G. r_coset G H x = l_coset G x H)" |
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|
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syntax (xsymbols) |
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normal :: "['a set, ('a,'b) monoid_scheme] => bool" (infixl "\<lhd>" 60) |
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subsection {*Lemmas Using Locale Constants*} |
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lemma (in group) r_coset_eq: "H #> a = {b . \<exists>h\<in>H. h \<otimes> a = b}" |
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by (auto simp add: r_coset_def) |
|
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lemma (in group) l_coset_eq: "a <# H = {b . \<exists>h\<in>H. a \<otimes> h = b}" |
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by (auto simp add: l_coset_def) |
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lemma (in group) setrcos_eq: "rcosets G H = {C . \<exists>a\<in>carrier G. C = H #> a}" |
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by (auto simp add: rcosets_def) |
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lemma (in group) set_mult_eq: "H <#> K = {c . \<exists>h\<in>H. \<exists>k\<in>K. c = h \<otimes> k}" |
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by (simp add: set_mult_def image_def) |
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lemma (in group) coset_mult_assoc: |
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"[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |] |
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==> (M #> g) #> h = M #> (g \<otimes> h)" |
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by (force simp add: r_coset_def m_assoc) |
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lemma (in group) coset_mult_one [simp]: "M \<subseteq> carrier G ==> M #> \<one> = M" |
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by (force simp add: r_coset_def) |
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lemma (in group) coset_mult_inv1: |
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"[| M #> (x \<otimes> (inv y)) = M; x \<in> carrier G ; y \<in> carrier G; |
14747 | 60 |
M \<subseteq> carrier G |] ==> M #> x = M #> y" |
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apply (erule subst [of concl: "%z. M #> x = z #> y"]) |
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apply (simp add: coset_mult_assoc m_assoc) |
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done |
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|
14747 | 65 |
lemma (in group) coset_mult_inv2: |
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"[| M #> x = M #> y; x \<in> carrier G; y \<in> carrier G; M \<subseteq> carrier G |] |
|
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==> M #> (x \<otimes> (inv y)) = M " |
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apply (simp add: coset_mult_assoc [symmetric]) |
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apply (simp add: coset_mult_assoc) |
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done |
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|
14747 | 72 |
lemma (in group) coset_join1: |
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"[| H #> x = H; x \<in> carrier G; subgroup H G |] ==> x \<in> H" |
|
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apply (erule subst) |
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75 |
apply (simp add: r_coset_eq) |
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apply (blast intro: l_one subgroup.one_closed) |
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77 |
done |
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|
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lemma (in group) solve_equation: |
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"\<lbrakk>subgroup H G; x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. h \<otimes> x = y" |
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apply (rule bexI [of _ "y \<otimes> (inv x)"]) |
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apply (auto simp add: subgroup.m_closed subgroup.m_inv_closed m_assoc |
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subgroup.subset [THEN subsetD]) |
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84 |
done |
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85 |
|
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lemma (in group) coset_join2: |
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"[| x \<in> carrier G; subgroup H G; x\<in>H |] ==> H #> x = H" |
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by (force simp add: subgroup.m_closed r_coset_eq solve_equation) |
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89 |
|
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lemma (in group) r_coset_subset_G: |
91 |
"[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H #> x \<subseteq> carrier G" |
|
92 |
by (auto simp add: r_coset_def) |
|
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93 |
|
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lemma (in group) rcosI: |
95 |
"[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<otimes> x \<in> H #> x" |
|
96 |
by (auto simp add: r_coset_def) |
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97 |
|
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lemma (in group) setrcosI: |
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"[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H #> x \<in> rcosets G H" |
|
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by (auto simp add: setrcos_eq) |
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101 |
|
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102 |
|
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text{*Really needed?*} |
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lemma (in group) transpose_inv: |
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"[| x \<otimes> y = z; x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] |
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==> (inv x) \<otimes> z = y" |
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107 |
by (force simp add: m_assoc [symmetric]) |
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108 |
|
14747 | 109 |
lemma (in group) repr_independence: |
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"[| y \<in> H #> x; x \<in> carrier G; subgroup H G |] ==> H #> x = H #> y" |
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by (auto simp add: r_coset_eq m_assoc [symmetric] |
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112 |
subgroup.subset [THEN subsetD] |
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113 |
subgroup.m_closed solve_equation) |
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114 |
|
14747 | 115 |
lemma (in group) rcos_self: "[| x \<in> carrier G; subgroup H G |] ==> x \<in> H #> x" |
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apply (simp add: r_coset_eq) |
14666 | 117 |
apply (blast intro: l_one subgroup.subset [THEN subsetD] |
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118 |
subgroup.one_closed) |
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119 |
done |
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120 |
|
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121 |
|
14666 | 122 |
subsection {* Normal subgroups *} |
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123 |
|
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lemma normal_imp_subgroup: "H <| G ==> subgroup H G" |
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125 |
by (simp add: normal_def) |
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126 |
|
14747 | 127 |
lemma (in group) normal_inv_op_closed1: |
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128 |
"\<lbrakk>H \<lhd> G; x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (inv x) \<otimes> h \<otimes> x \<in> H" |
14747 | 129 |
apply (auto simp add: l_coset_def r_coset_def normal_def) |
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apply (drule bspec, assumption) |
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apply (drule equalityD1 [THEN subsetD], blast, clarify) |
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132 |
apply (simp add: m_assoc subgroup.subset [THEN subsetD]) |
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133 |
apply (simp add: m_assoc [symmetric] subgroup.subset [THEN subsetD]) |
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134 |
done |
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135 |
|
14747 | 136 |
lemma (in group) normal_inv_op_closed2: |
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137 |
"\<lbrakk>H \<lhd> G; x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> (inv x) \<in> H" |
14666 | 138 |
apply (drule normal_inv_op_closed1 [of H "(inv x)"]) |
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139 |
apply auto |
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140 |
done |
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141 |
|
14747 | 142 |
text{*Alternative characterization of normal subgroups*} |
143 |
lemma (in group) normal_inv_iff: |
|
144 |
"(N \<lhd> G) = |
|
145 |
(subgroup N G & (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<otimes> h \<otimes> (inv x) \<in> N))" |
|
146 |
(is "_ = ?rhs") |
|
147 |
proof |
|
148 |
assume N: "N \<lhd> G" |
|
149 |
show ?rhs |
|
150 |
by (blast intro: N normal_imp_subgroup normal_inv_op_closed2) |
|
151 |
next |
|
152 |
assume ?rhs |
|
153 |
hence sg: "subgroup N G" |
|
154 |
and closed: "!!x. x\<in>carrier G ==> \<forall>h\<in>N. x \<otimes> h \<otimes> inv x \<in> N" by auto |
|
155 |
hence sb: "N \<subseteq> carrier G" by (simp add: subgroup.subset) |
|
156 |
show "N \<lhd> G" |
|
157 |
proof (simp add: sg normal_def l_coset_def r_coset_def, clarify) |
|
158 |
fix x |
|
159 |
assume x: "x \<in> carrier G" |
|
160 |
show "(\<lambda>n. n \<otimes> x) ` N = op \<otimes> x ` N" |
|
161 |
proof |
|
162 |
show "(\<lambda>n. n \<otimes> x) ` N \<subseteq> op \<otimes> x ` N" |
|
163 |
proof clarify |
|
164 |
fix n |
|
165 |
assume n: "n \<in> N" |
|
166 |
show "n \<otimes> x \<in> op \<otimes> x ` N" |
|
167 |
proof |
|
168 |
show "n \<otimes> x = x \<otimes> (inv x \<otimes> n \<otimes> x)" |
|
169 |
by (simp add: x n m_assoc [symmetric] sb [THEN subsetD]) |
|
170 |
with closed [of "inv x"] |
|
171 |
show "inv x \<otimes> n \<otimes> x \<in> N" by (simp add: x n) |
|
172 |
qed |
|
173 |
qed |
|
174 |
next |
|
175 |
show "op \<otimes> x ` N \<subseteq> (\<lambda>n. n \<otimes> x) ` N" |
|
176 |
proof clarify |
|
177 |
fix n |
|
178 |
assume n: "n \<in> N" |
|
179 |
show "x \<otimes> n \<in> (\<lambda>n. n \<otimes> x) ` N" |
|
180 |
proof |
|
181 |
show "x \<otimes> n = (x \<otimes> n \<otimes> inv x) \<otimes> x" |
|
182 |
by (simp add: x n m_assoc sb [THEN subsetD]) |
|
183 |
show "x \<otimes> n \<otimes> inv x \<in> N" by (simp add: x n closed) |
|
184 |
qed |
|
185 |
qed |
|
186 |
qed |
|
187 |
qed |
|
188 |
qed |
|
13870
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paulson
parents:
diff
changeset
|
189 |
|
14747 | 190 |
lemma (in group) lcos_m_assoc: |
191 |
"[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |] |
|
192 |
==> g <# (h <# M) = (g \<otimes> h) <# M" |
|
193 |
by (force simp add: l_coset_def m_assoc) |
|
13870
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moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
194 |
|
14747 | 195 |
lemma (in group) lcos_mult_one: "M \<subseteq> carrier G ==> \<one> <# M = M" |
196 |
by (force simp add: l_coset_def) |
|
13870
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moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
197 |
|
14747 | 198 |
lemma (in group) l_coset_subset_G: |
199 |
"[| H \<subseteq> carrier G; x \<in> carrier G |] ==> x <# H \<subseteq> carrier G" |
|
200 |
by (auto simp add: l_coset_def subsetD) |
|
201 |
||
202 |
lemma (in group) l_coset_swap: |
|
14530 | 203 |
"[| y \<in> x <# H; x \<in> carrier G; subgroup H G |] ==> x \<in> y <# H" |
204 |
proof (simp add: l_coset_eq) |
|
14666 | 205 |
assume "\<exists>h\<in>H. x \<otimes> h = y" |
206 |
and x: "x \<in> carrier G" |
|
14530 | 207 |
and sb: "subgroup H G" |
208 |
then obtain h' where h': "h' \<in> H & x \<otimes> h' = y" by blast |
|
209 |
show "\<exists>h\<in>H. y \<otimes> h = x" |
|
210 |
proof |
|
211 |
show "y \<otimes> inv h' = x" using h' x sb |
|
212 |
by (auto simp add: m_assoc subgroup.subset [THEN subsetD]) |
|
213 |
show "inv h' \<in> H" using h' sb |
|
214 |
by (auto simp add: subgroup.subset [THEN subsetD] subgroup.m_inv_closed) |
|
215 |
qed |
|
216 |
qed |
|
217 |
||
14747 | 218 |
lemma (in group) l_coset_carrier: |
14530 | 219 |
"[| y \<in> x <# H; x \<in> carrier G; subgroup H G |] ==> y \<in> carrier G" |
14747 | 220 |
by (auto simp add: l_coset_def m_assoc |
14530 | 221 |
subgroup.subset [THEN subsetD] subgroup.m_closed) |
222 |
||
14747 | 223 |
lemma (in group) l_repr_imp_subset: |
14666 | 224 |
assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G" |
14530 | 225 |
shows "y <# H \<subseteq> x <# H" |
226 |
proof - |
|
227 |
from y |
|
14747 | 228 |
obtain h' where "h' \<in> H" "x \<otimes> h' = y" by (auto simp add: l_coset_def) |
14530 | 229 |
thus ?thesis using x sb |
14747 | 230 |
by (auto simp add: l_coset_def m_assoc |
14530 | 231 |
subgroup.subset [THEN subsetD] subgroup.m_closed) |
232 |
qed |
|
233 |
||
14747 | 234 |
lemma (in group) l_repr_independence: |
14666 | 235 |
assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G" |
14530 | 236 |
shows "x <# H = y <# H" |
14666 | 237 |
proof |
14530 | 238 |
show "x <# H \<subseteq> y <# H" |
14666 | 239 |
by (rule l_repr_imp_subset, |
14530 | 240 |
(blast intro: l_coset_swap l_coset_carrier y x sb)+) |
14666 | 241 |
show "y <# H \<subseteq> x <# H" by (rule l_repr_imp_subset [OF y x sb]) |
14530 | 242 |
qed |
13870
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moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
243 |
|
14747 | 244 |
lemma (in group) setmult_subset_G: |
245 |
"[| H \<subseteq> carrier G; K \<subseteq> carrier G |] ==> H <#> K \<subseteq> carrier G" |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
246 |
by (auto simp add: set_mult_eq subsetD) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
247 |
|
14747 | 248 |
lemma (in group) subgroup_mult_id: "subgroup H G ==> H <#> H = H" |
13870
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moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
249 |
apply (auto simp add: subgroup.m_closed set_mult_eq Sigma_def image_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
250 |
apply (rule_tac x = x in bexI) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
251 |
apply (rule bexI [of _ "\<one>"]) |
14666 | 252 |
apply (auto simp add: subgroup.m_closed subgroup.one_closed |
13870
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moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
253 |
r_one subgroup.subset [THEN subsetD]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
254 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
255 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
256 |
|
14747 | 257 |
subsubsection {* Set of inverses of an @{text r_coset}. *} |
14666 | 258 |
|
14747 | 259 |
lemma (in group) rcos_inv: |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
260 |
assumes normalHG: "H <| G" |
14747 | 261 |
and x: "x \<in> carrier G" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
262 |
shows "set_inv G (H #> x) = H #> (inv x)" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
263 |
proof - |
14747 | 264 |
have H_subset: "H \<subseteq> carrier G" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
265 |
by (rule subgroup.subset [OF normal_imp_subgroup, OF normalHG]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
266 |
show ?thesis |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
267 |
proof (auto simp add: r_coset_eq image_def set_inv_def) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
268 |
fix h |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
269 |
assume "h \<in> H" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
270 |
hence "((inv x) \<otimes> (inv h) \<otimes> x) \<otimes> inv x = inv (h \<otimes> x)" |
14747 | 271 |
by (simp add: x m_assoc inv_mult_group H_subset [THEN subsetD]) |
14666 | 272 |
thus "\<exists>j\<in>H. j \<otimes> inv x = inv (h \<otimes> x)" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
273 |
using prems |
14666 | 274 |
by (blast intro: normal_inv_op_closed1 normal_imp_subgroup |
275 |
subgroup.m_inv_closed) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
276 |
next |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
277 |
fix h |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
278 |
assume "h \<in> H" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
279 |
hence eq: "(x \<otimes> (inv h) \<otimes> (inv x)) \<otimes> x = x \<otimes> inv h" |
14747 | 280 |
by (simp add: x m_assoc H_subset [THEN subsetD]) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
281 |
hence "(\<exists>j\<in>H. j \<otimes> x = inv (h \<otimes> (inv x))) \<and> h \<otimes> inv x = inv (inv (h \<otimes> (inv x)))" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
282 |
using prems |
14666 | 283 |
by (simp add: m_assoc inv_mult_group H_subset [THEN subsetD], |
284 |
blast intro: eq normal_inv_op_closed2 normal_imp_subgroup |
|
285 |
subgroup.m_inv_closed) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
286 |
thus "\<exists>y. (\<exists>h\<in>H. h \<otimes> x = y) \<and> h \<otimes> inv x = inv y" .. |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
287 |
qed |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
288 |
qed |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
289 |
|
14747 | 290 |
lemma (in group) rcos_inv2: |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
291 |
"[| H <| G; K \<in> rcosets G H; x \<in> K |] ==> set_inv G K = H #> (inv x)" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
292 |
apply (simp add: setrcos_eq, clarify) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
293 |
apply (subgoal_tac "x : carrier G") |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
294 |
prefer 2 |
14666 | 295 |
apply (blast dest: r_coset_subset_G subgroup.subset normal_imp_subgroup) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
296 |
apply (drule repr_independence) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
297 |
apply assumption |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
298 |
apply (erule normal_imp_subgroup) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
299 |
apply (simp add: rcos_inv) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
300 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
301 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
302 |
|
14747 | 303 |
subsubsection {* Some rules for @{text "<#>"} with @{text "#>"} or @{text "<#"}. *} |
14666 | 304 |
|
14747 | 305 |
lemma (in group) setmult_rcos_assoc: |
306 |
"[| H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G |] |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
307 |
==> H <#> (K #> x) = (H <#> K) #> x" |
14747 | 308 |
apply (auto simp add: r_coset_def set_mult_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
309 |
apply (force simp add: m_assoc)+ |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
310 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
311 |
|
14747 | 312 |
lemma (in group) rcos_assoc_lcos: |
313 |
"[| H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G |] |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
314 |
==> (H #> x) <#> K = H <#> (x <# K)" |
14747 | 315 |
apply (auto simp add: r_coset_def l_coset_def set_mult_def Sigma_def image_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
316 |
apply (force intro!: exI bexI simp add: m_assoc)+ |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
317 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
318 |
|
14747 | 319 |
lemma (in group) rcos_mult_step1: |
14666 | 320 |
"[| H <| G; x \<in> carrier G; y \<in> carrier G |] |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
321 |
==> (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
322 |
by (simp add: setmult_rcos_assoc normal_imp_subgroup [THEN subgroup.subset] |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
323 |
r_coset_subset_G l_coset_subset_G rcos_assoc_lcos) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
324 |
|
14747 | 325 |
lemma (in group) rcos_mult_step2: |
14666 | 326 |
"[| H <| G; x \<in> carrier G; y \<in> carrier G |] |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
327 |
==> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y" |
14747 | 328 |
by (simp add: normal_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
329 |
|
14747 | 330 |
lemma (in group) rcos_mult_step3: |
14666 | 331 |
"[| H <| G; x \<in> carrier G; y \<in> carrier G |] |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
332 |
==> (H <#> (H #> x)) #> y = H #> (x \<otimes> y)" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
333 |
by (simp add: setmult_rcos_assoc r_coset_subset_G coset_mult_assoc |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
334 |
setmult_subset_G subgroup_mult_id |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
335 |
subgroup.subset normal_imp_subgroup) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
336 |
|
14747 | 337 |
lemma (in group) rcos_sum: |
14666 | 338 |
"[| H <| G; x \<in> carrier G; y \<in> carrier G |] |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
339 |
==> (H #> x) <#> (H #> y) = H #> (x \<otimes> y)" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
340 |
by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
341 |
|
14747 | 342 |
lemma (in group) setrcos_mult_eq: "[|H <| G; M \<in> rcosets G H|] ==> H <#> M = M" |
14666 | 343 |
-- {* generalizes @{text subgroup_mult_id} *} |
344 |
by (auto simp add: setrcos_eq normal_imp_subgroup subgroup.subset |
|
345 |
setmult_rcos_assoc subgroup_mult_id) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
346 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
347 |
|
14666 | 348 |
subsection {*Lemmas Leading to Lagrange's Theorem *} |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
349 |
|
14747 | 350 |
lemma (in group) setrcos_part_G: "subgroup H G ==> \<Union>rcosets G H = carrier G" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
351 |
apply (rule equalityI) |
14666 | 352 |
apply (force simp add: subgroup.subset [THEN subsetD] |
14747 | 353 |
setrcos_eq r_coset_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
354 |
apply (auto simp add: setrcos_eq subgroup.subset rcos_self) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
355 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
356 |
|
14747 | 357 |
lemma (in group) cosets_finite: |
358 |
"[| c \<in> rcosets G H; H \<subseteq> carrier G; finite (carrier G) |] ==> finite c" |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
359 |
apply (auto simp add: setrcos_eq) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
360 |
apply (simp (no_asm_simp) add: r_coset_subset_G [THEN finite_subset]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
361 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
362 |
|
14747 | 363 |
text{*The next two lemmas support the proof of @{text card_cosets_equal}.*} |
364 |
lemma (in group) inj_on_f: |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
365 |
"[|H \<subseteq> carrier G; a \<in> carrier G|] ==> inj_on (\<lambda>y. y \<otimes> inv a) (H #> a)" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
366 |
apply (rule inj_onI) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
367 |
apply (subgoal_tac "x \<in> carrier G & y \<in> carrier G") |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
368 |
prefer 2 apply (blast intro: r_coset_subset_G [THEN subsetD]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
369 |
apply (simp add: subsetD) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
370 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
371 |
|
14747 | 372 |
lemma (in group) inj_on_g: |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
373 |
"[|H \<subseteq> carrier G; a \<in> carrier G|] ==> inj_on (\<lambda>y. y \<otimes> a) H" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
374 |
by (force simp add: inj_on_def subsetD) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
375 |
|
14747 | 376 |
lemma (in group) card_cosets_equal: |
377 |
"[| c \<in> rcosets G H; H \<subseteq> carrier G; finite(carrier G) |] |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
378 |
==> card c = card H" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
379 |
apply (auto simp add: setrcos_eq) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
380 |
apply (rule card_bij_eq) |
14666 | 381 |
apply (rule inj_on_f, assumption+) |
14747 | 382 |
apply (force simp add: m_assoc subsetD r_coset_def) |
14666 | 383 |
apply (rule inj_on_g, assumption+) |
14747 | 384 |
apply (force simp add: m_assoc subsetD r_coset_def) |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
385 |
txt{*The sets @{term "H #> a"} and @{term "H"} are finite.*} |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
386 |
apply (simp add: r_coset_subset_G [THEN finite_subset]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
387 |
apply (blast intro: finite_subset) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
388 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
389 |
|
14747 | 390 |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
391 |
subsection{*Two distinct right cosets are disjoint*} |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
392 |
|
14747 | 393 |
lemma (in group) rcos_equation: |
14666 | 394 |
"[|subgroup H G; a \<in> carrier G; b \<in> carrier G; ha \<otimes> a = h \<otimes> b; |
395 |
h \<in> H; ha \<in> H; hb \<in> H|] |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
396 |
==> \<exists>h\<in>H. h \<otimes> b = hb \<otimes> a" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
397 |
apply (rule bexI [of _"hb \<otimes> ((inv ha) \<otimes> h)"]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
398 |
apply (simp add: m_assoc transpose_inv subgroup.subset [THEN subsetD]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
399 |
apply (simp add: subgroup.m_closed subgroup.m_inv_closed) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
400 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
401 |
|
14747 | 402 |
lemma (in group) rcos_disjoint: |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
403 |
"[|subgroup H G; a \<in> rcosets G H; b \<in> rcosets G H; a\<noteq>b|] ==> a \<inter> b = {}" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
404 |
apply (simp add: setrcos_eq r_coset_eq) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
405 |
apply (blast intro: rcos_equation sym) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
406 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
407 |
|
14747 | 408 |
lemma (in group) setrcos_subset_PowG: |
409 |
"subgroup H G ==> rcosets G H \<subseteq> Pow(carrier G)" |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
410 |
apply (simp add: setrcos_eq) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
411 |
apply (blast dest: r_coset_subset_G subgroup.subset) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
412 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
413 |
|
14747 | 414 |
subsection {*Quotient Groups: Factorization of a Group*} |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
415 |
|
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
416 |
constdefs |
13936 | 417 |
FactGroup :: "[('a,'b) monoid_scheme, 'a set] => ('a set) monoid" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
418 |
(infixl "Mod" 60) |
14747 | 419 |
--{*Actually defined for groups rather than monoids*} |
14666 | 420 |
"FactGroup G H == |
421 |
(| carrier = rcosets G H, |
|
422 |
mult = (%X: rcosets G H. %Y: rcosets G H. set_mult G X Y), |
|
14747 | 423 |
one = H |)" |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
424 |
|
14747 | 425 |
|
426 |
lemma (in group) setmult_closed: |
|
14666 | 427 |
"[| H <| G; K1 \<in> rcosets G H; K2 \<in> rcosets G H |] |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
428 |
==> K1 <#> K2 \<in> rcosets G H" |
14666 | 429 |
by (auto simp add: normal_imp_subgroup [THEN subgroup.subset] |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
430 |
rcos_sum setrcos_eq) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
431 |
|
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
432 |
lemma (in group) setinv_closed: |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
433 |
"[| H <| G; K \<in> rcosets G H |] ==> set_inv G K \<in> rcosets G H" |
14747 | 434 |
by (auto simp add: normal_imp_subgroup |
435 |
subgroup.subset rcos_inv |
|
436 |
setrcos_eq) |
|
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
437 |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
438 |
(* |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
439 |
The old version is no longer valid: "group G" has to be an explicit premise. |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
440 |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
441 |
lemma setinv_closed: |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
442 |
"[| H <| G; K \<in> rcosets G H |] ==> set_inv G K \<in> rcosets G H" |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
443 |
by (auto simp add: normal_imp_subgroup |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
444 |
subgroup.subset coset.rcos_inv coset.setrcos_eq) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
445 |
*) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
446 |
|
14747 | 447 |
lemma (in group) setrcos_assoc: |
14666 | 448 |
"[|H <| G; M1 \<in> rcosets G H; M2 \<in> rcosets G H; M3 \<in> rcosets G H|] |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
449 |
==> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)" |
14666 | 450 |
by (auto simp add: setrcos_eq rcos_sum normal_imp_subgroup |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
451 |
subgroup.subset m_assoc) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
452 |
|
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
453 |
lemma (in group) subgroup_in_rcosets: |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
454 |
"subgroup H G ==> H \<in> rcosets G H" |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
455 |
proof - |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
456 |
assume sub: "subgroup H G" |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
457 |
have "r_coset G H \<one> = H" |
14747 | 458 |
by (rule coset_join2) |
459 |
(auto intro: sub subgroup.one_closed) |
|
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
460 |
then show ?thesis |
14747 | 461 |
by (auto simp add: setrcos_eq) |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
462 |
qed |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
463 |
|
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
464 |
(* |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
465 |
lemma subgroup_in_rcosets: |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
466 |
"subgroup H G ==> H \<in> rcosets G H" |
14666 | 467 |
apply (frule subgroup_imp_coset) |
468 |
apply (frule subgroup_imp_group) |
|
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
469 |
apply (simp add: coset.setrcos_eq) |
14666 | 470 |
apply (blast del: equalityI |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
471 |
intro!: group.subgroup.one_closed group.one_closed |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
472 |
coset.coset_join2 [symmetric]) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
473 |
done |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
474 |
*) |
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
475 |
|
14747 | 476 |
lemma (in group) setrcos_inv_mult_group_eq: |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
477 |
"[|H <| G; M \<in> rcosets G H|] ==> set_inv G M <#> M = H" |
14666 | 478 |
by (auto simp add: setrcos_eq rcos_inv rcos_sum normal_imp_subgroup |
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
479 |
subgroup.subset) |
13940 | 480 |
(* |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
481 |
lemma (in group) factorgroup_is_magma: |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
482 |
"H <| G ==> magma (G Mod H)" |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
483 |
by rule (simp add: FactGroup_def coset.setmult_closed [OF is_coset]) |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
484 |
|
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
485 |
lemma (in group) factorgroup_semigroup_axioms: |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
486 |
"H <| G ==> semigroup_axioms (G Mod H)" |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
487 |
by rule (simp add: FactGroup_def coset.setrcos_assoc [OF is_coset] |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
488 |
coset.setmult_closed [OF is_coset]) |
13940 | 489 |
*) |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
490 |
theorem (in group) factorgroup_is_group: |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
491 |
"H <| G ==> group (G Mod H)" |
14666 | 492 |
apply (simp add: FactGroup_def) |
13936 | 493 |
apply (rule groupI) |
14747 | 494 |
apply (simp add: setmult_closed) |
13936 | 495 |
apply (simp add: normal_imp_subgroup subgroup_in_rcosets) |
14747 | 496 |
apply (simp add: restrictI setmult_closed setrcos_assoc) |
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
497 |
apply (simp add: normal_imp_subgroup |
14747 | 498 |
subgroup_in_rcosets setrcos_mult_eq) |
499 |
apply (auto dest: setrcos_inv_mult_group_eq simp add: setinv_closed) |
|
13889
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
500 |
done |
6676ac2527fa
Fixed Coset.thy (proved theorem factorgroup_is_group).
ballarin
parents:
13870
diff
changeset
|
501 |
|
14747 | 502 |
lemma (in group) inv_FactGroup: |
503 |
"N <| G ==> X \<in> carrier (G Mod N) ==> inv\<^bsub>G Mod N\<^esub> X = set_inv G X" |
|
504 |
apply (rule group.inv_equality [OF factorgroup_is_group]) |
|
505 |
apply (simp_all add: FactGroup_def setinv_closed |
|
506 |
setrcos_inv_mult_group_eq group.intro [OF prems]) |
|
507 |
done |
|
508 |
||
509 |
text{*The coset map is a homomorphism from @{term G} to the quotient group |
|
510 |
@{term "G Mod N"}*} |
|
511 |
lemma (in group) r_coset_hom_Mod: |
|
512 |
assumes N: "N <| G" |
|
513 |
shows "(r_coset G N) \<in> hom G (G Mod N)" |
|
514 |
by (simp add: FactGroup_def rcosets_def Pi_def hom_def |
|
515 |
rcos_sum group.intro prems) |
|
516 |
||
13870
cf947d1ec5ff
moved Exponent, Coset, Sylow from GroupTheory to Algebra, converting them
paulson
parents:
diff
changeset
|
517 |
end |