src/HOL/Algebra/SimpleGroups.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Algebra/SimpleGroups.thy	Mon Feb 27 17:09:59 2023 +0000
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+(*  Title:      Simple Groups
+    Author:     Jakob von Raumer, Karlsruhe Institute of Technology
+    Maintainer: Jakob von Raumer <[e-mail]>
+*)
+
+theory SimpleGroups
+imports Coset "HOL-Computational_Algebra.Primes"
+begin
+
+section \<open>Simple Groups\<close>
+
+locale simple_group = group +
+  assumes order_gt_one: "order G > 1"
+  assumes no_real_normal_subgroup: "\<And>H. H \<lhd> G \<Longrightarrow> (H = carrier G \<or> H = {\<one>})"
+
+lemma (in simple_group) is_simple_group: "simple_group G" 
+  by (rule simple_group_axioms)
+
+text \<open>Simple groups are non-trivial.\<close>
+
+lemma (in simple_group) simple_not_triv: "carrier G \<noteq> {\<one>}" 
+  using order_gt_one unfolding order_def by auto
+
+text \<open>Every group of prime order is simple\<close>
+
+lemma (in group) prime_order_simple:
+  assumes prime: "prime (order G)"
+  shows "simple_group G"
+proof
+  from prime show "1 < order G" 
+    unfolding prime_nat_iff by auto
+next
+  fix H
+  assume "H \<lhd> G"
+  hence HG: "subgroup H G" unfolding normal_def by simp
+  hence "card H dvd order G"
+    by (metis dvd_triv_right lagrange)
+  with prime have "card H = 1 \<or> card H = order G" 
+    unfolding prime_nat_iff by simp
+  thus "H = carrier G \<or> H = {\<one>}"
+  proof
+    assume "card H = 1"
+    moreover from HG have "\<one> \<in> H" by (metis subgroup.one_closed)
+    ultimately show ?thesis by (auto simp: card_Suc_eq)
+  next
+    assume "card H = order G"
+    moreover from HG have "H \<subseteq> carrier G" unfolding subgroup_def by simp
+    moreover from prime have "finite (carrier G)"
+      using order_gt_0_iff_finite by force
+    ultimately show ?thesis 
+      unfolding order_def by (metis card_subset_eq)
+  qed
+qed
+
+text \<open>Being simple is a property that is preserved by isomorphisms.\<close>
+
+lemma (in simple_group) iso_simple:
+  assumes H: "group H"
+  assumes iso: "\<phi> \<in> iso G H"
+  shows "simple_group H"
+unfolding simple_group_def simple_group_axioms_def 
+proof (intro conjI strip H)
+  from iso have "order G = order H" unfolding iso_def order_def using bij_betw_same_card by auto
+  with order_gt_one show "1 < order H" by simp
+next
+  have inv_iso: "(inv_into (carrier G) \<phi>) \<in> iso H G" using iso
+    by (simp add: iso_set_sym)    
+  fix N
+  assume NH: "N \<lhd> H" 
+  then interpret Nnormal: normal N H by simp
+  define M where "M = (inv_into (carrier G) \<phi>) ` N"
+  hence MG: "M \<lhd> G" 
+    using inv_iso NH H by (metis is_group iso_normal_subgroup)
+  have surj: "\<phi> ` carrier G = carrier H" 
+    using iso unfolding iso_def bij_betw_def by simp
+  hence MN: "\<phi> ` M = N" 
+    unfolding M_def using Nnormal.subset image_inv_into_cancel by metis
+  then have "N = {\<one>\<^bsub>H\<^esub>}" if "M = {\<one>}"
+    using Nnormal.subgroup_axioms subgroup.one_closed that by force
+  then show "N = carrier H \<or> N = {\<one>\<^bsub>H\<^esub>}"
+    by (metis MG MN no_real_normal_subgroup surj)
+qed
+
+text \<open>As a corollary of this: Factorizing a group by itself does not result in a simple group!\<close>
+
+lemma (in group) self_factor_not_simple: "\<not> simple_group (G Mod (carrier G))"
+proof
+  assume assm: "simple_group (G Mod (carrier G))"
+  with self_factor_iso simple_group.iso_simple have "simple_group (G\<lparr>carrier := {\<one>}\<rparr>)"
+    using subgroup_imp_group triv_subgroup by blast
+  thus False 
+    using simple_group.simple_not_triv by force
+qed
+
+end