(* Title: Simple Groups
Author: Jakob von Raumer, Karlsruhe Institute of Technology
Maintainer: Jakob von Raumer <jakob.raumer@student.kit.edu>
*)
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