# Theory Group_Action

```(*  Title:      HOL/Algebra/Group_Action.thy
Author:     Paulo Emílio de Vilhena
*)

theory Group_Action
imports Bij Coset Congruence
begin

section ‹Group Actions›

locale group_action =
fixes G (structure) and E and φ
assumes group_hom: "group_hom G (BijGroup E) φ"

definition
orbit :: "[_, 'a ⇒ 'b ⇒ 'b, 'b] ⇒ 'b set"
where "orbit G φ x = {(φ g) x | g. g ∈ carrier G}"

definition
orbits :: "[_, 'b set, 'a ⇒ 'b ⇒ 'b] ⇒ ('b set) set"
where "orbits G E φ = {orbit G φ x | x. x ∈ E}"

definition
stabilizer :: "[_, 'a ⇒ 'b ⇒ 'b, 'b] ⇒ 'a set"
where "stabilizer G φ x = {g ∈ carrier G. (φ g) x = x}"

definition
invariants :: "['b set, 'a ⇒ 'b ⇒ 'b, 'a] ⇒ 'b set"
where "invariants E φ g = {x ∈ E. (φ g) x = x}"

definition
normalizer :: "[_, 'a set] ⇒ 'a set"
where "normalizer G H =
stabilizer G (λg. λH ∈ {H. H ⊆ carrier G}. g <#⇘G⇙ H #>⇘G⇙ (inv⇘G⇙ g)) H"

locale faithful_action = group_action +
assumes faithful: "inj_on φ (carrier G)"

locale transitive_action = group_action +
assumes unique_orbit: "⟦ x ∈ E; y ∈ E ⟧ ⟹ ∃g ∈ carrier G. (φ g) x = y"

subsection ‹Prelimineries›

text ‹Some simple lemmas to make group action's properties more explicit›

lemma (in group_action) id_eq_one: "(λx ∈ E. x) = φ 𝟭"
by (metis BijGroup_def group_hom group_hom.hom_one select_convs(2))

lemma (in group_action) bij_prop0:
"⋀ g. g ∈ carrier G ⟹ (φ g) ∈ Bij E"
by (metis BijGroup_def group_hom group_hom.hom_closed partial_object.select_convs(1))

lemma (in group_action) surj_prop:
"⋀ g. g ∈ carrier G ⟹ (φ g) ` E = E"
using bij_prop0 by (simp add: Bij_def bij_betw_def)

lemma (in group_action) inj_prop:
"⋀ g. g ∈ carrier G ⟹ inj_on (φ g) E"
using bij_prop0 by (simp add: Bij_def bij_betw_def)

lemma (in group_action) bij_prop1:
"⋀ g y. ⟦ g ∈ carrier G; y ∈ E ⟧ ⟹  ∃!x ∈ E. (φ g) x = y"
proof -
fix g y assume "g ∈ carrier G" "y ∈ E"
hence "∃x ∈ E. (φ g) x = y"
using surj_prop by force
moreover have "⋀ x1 x2. ⟦ x1 ∈ E; x2 ∈ E ⟧ ⟹ (φ g) x1 = (φ g) x2 ⟹ x1 = x2"
using inj_prop by (meson ‹g ∈ carrier G› inj_on_eq_iff)
ultimately show "∃!x ∈ E. (φ g) x = y"
by blast
qed

lemma (in group_action) composition_rule:
assumes "x ∈ E" "g1 ∈ carrier G" "g2 ∈ carrier G"
shows "φ (g1 ⊗ g2) x = (φ g1) (φ g2 x)"
proof -
have "φ (g1 ⊗ g2) x = ((φ g1) ⊗⇘BijGroup E⇙ (φ g2)) x"
using assms(2) assms(3) group_hom group_hom.hom_mult by fastforce
also have " ... = (compose E (φ g1) (φ g2)) x"
unfolding BijGroup_def by (simp add: assms bij_prop0)
finally show "φ (g1 ⊗ g2) x = (φ g1) (φ g2 x)"
qed

lemma (in group_action) element_image:
assumes "g ∈ carrier G" and "x ∈ E" and "(φ g) x = y"
shows "y ∈ E"
using surj_prop assms by blast

subsection ‹Orbits›

text‹We prove here that orbits form an equivalence relation›

lemma (in group_action) orbit_sym_aux:
assumes "g ∈ carrier G"
and "x ∈ E"
and "(φ g) x = y"
shows "(φ (inv g)) y = x"
proof -
interpret group G
using group_hom group_hom.axioms(1) by auto
have "y ∈ E"
using element_image assms by simp
have "inv g ∈ carrier G"

have "(φ (inv g)) y = (φ (inv g)) ((φ g) x)"
using assms(3) by simp
also have " ... = compose E (φ (inv g)) (φ g) x"
also have " ... = ((φ (inv g)) ⊗⇘BijGroup E⇙ (φ g)) x"
by (simp add: BijGroup_def assms(1) bij_prop0)
also have " ... = (φ ((inv g) ⊗ g)) x"
by (metis ‹inv g ∈ carrier G› assms(1) group_hom group_hom.hom_mult)
finally show "(φ (inv g)) y = x"
by (metis assms(1) assms(2) id_eq_one l_inv restrict_apply)
qed

lemma (in group_action) orbit_refl:
"x ∈ E ⟹ x ∈ orbit G φ x"
proof -
assume "x ∈ E" hence "(φ 𝟭) x = x"
using id_eq_one by (metis restrict_apply')
thus "x ∈ orbit G φ x" unfolding orbit_def
using group.is_monoid group_hom group_hom.axioms(1) by force
qed

lemma (in group_action) orbit_sym:
assumes "x ∈ E" and "y ∈ E" and "y ∈ orbit G φ x"
shows "x ∈ orbit G φ y"
proof -
have "∃ g ∈ carrier G. (φ g) x = y"
using assms by (auto simp: orbit_def)
then obtain g where g: "g ∈ carrier G ∧ (φ g) x = y" by blast
hence "(φ (inv g)) y = x"
using orbit_sym_aux by (simp add: assms(1))
thus ?thesis
using g group_hom group_hom.axioms(1) orbit_def by fastforce
qed

lemma (in group_action) orbit_trans:
assumes "x ∈ E" "y ∈ E" "z ∈ E"
and "y ∈ orbit G φ x" "z ∈ orbit G φ y"
shows "z ∈ orbit G φ x"
proof -
interpret group G
using group_hom group_hom.axioms(1) by auto
obtain g1 where g1: "g1 ∈ carrier G ∧ (φ g1) x = y"
using assms by (auto simp: orbit_def)
obtain g2 where g2: "g2 ∈ carrier G ∧ (φ g2) y = z"
using assms by (auto simp: orbit_def)
have "(φ (g2 ⊗ g1)) x = ((φ g2) ⊗⇘BijGroup E⇙ (φ g1)) x"
using g1 g2 group_hom group_hom.hom_mult by fastforce
also have " ... = (φ g2) ((φ g1) x)"
using composition_rule assms(1) calculation g1 g2 by auto
finally have "(φ (g2 ⊗ g1)) x = z"
thus ?thesis
using g1 g2 orbit_def by force
qed

lemma (in group_action) orbits_as_classes:
"classes⇘⦇ carrier = E, eq = λx. λy. y ∈ orbit G φ x ⦈⇙ = orbits G E φ"
unfolding eq_classes_def eq_class_of_def orbits_def orbit_def
using element_image by auto

theorem (in group_action) orbit_partition:
"partition E (orbits G E φ)"
proof -
have "equivalence ⦇ carrier = E, eq = λx. λy. y ∈ orbit G φ x ⦈"
unfolding equivalence_def apply simp
using orbit_refl orbit_sym orbit_trans by blast
thus ?thesis using equivalence.partition_from_equivalence orbits_as_classes by fastforce
qed

corollary (in group_action) orbits_coverture:
"⋃ (orbits G E φ) = E"
using partition.partition_coverture[OF orbit_partition] by simp

corollary (in group_action) disjoint_union:
assumes "orb1 ∈ (orbits G E φ)" "orb2 ∈ (orbits G E φ)"
shows "(orb1 = orb2) ∨ (orb1 ∩ orb2) = {}"
using partition.disjoint_union[OF orbit_partition] assms by auto

corollary (in group_action) disjoint_sum:
assumes "finite E"
shows "(∑orb∈(orbits G E φ). ∑x∈orb. f x) = (∑x∈E. f x)"
using partition.disjoint_sum[OF orbit_partition] assms by auto

subsubsection ‹Transitive Actions›

text ‹Transitive actions have only one orbit›

lemma (in transitive_action) all_equivalent:
"⟦ x ∈ E; y ∈ E ⟧ ⟹ x .=⇘⦇carrier = E, eq = λx y. y ∈ orbit G φ x⦈⇙ y"
proof -
assume "x ∈ E" "y ∈ E"
hence "∃ g ∈ carrier G. (φ g) x = y"
using unique_orbit  by blast
hence "y ∈ orbit G φ x"
using orbit_def by fastforce
thus "x .=⇘⦇carrier = E, eq = λx y. y ∈ orbit G φ x⦈⇙ y" by simp
qed

proposition (in transitive_action) one_orbit:
assumes "E ≠ {}"
shows "card (orbits G E φ) = 1"
proof -
have "orbits G E φ ≠ {}"
using assms orbits_coverture by auto
moreover have "⋀ orb1 orb2. ⟦ orb1 ∈ (orbits G E φ); orb2 ∈ (orbits G E φ) ⟧ ⟹ orb1 = orb2"
proof -
fix orb1 orb2 assume orb1: "orb1 ∈ (orbits G E φ)"
and orb2: "orb2 ∈ (orbits G E φ)"
then obtain x y where x: "orb1 = orbit G φ x" and x_E: "x ∈ E"
and y: "orb2 = orbit G φ y" and y_E: "y ∈ E"
unfolding orbits_def by blast
hence "x ∈ orbit G φ y" using all_equivalent by auto
hence "orb1 ∩ orb2 ≠ {}" using x y x_E orbit_refl by auto
thus "orb1 = orb2" using disjoint_union[of orb1 orb2] orb1 orb2 by auto
qed
ultimately show "card (orbits G E φ) = 1"
by (meson is_singletonI' is_singleton_altdef)
qed

subsection ‹Stabilizers›

text ‹We show that stabilizers are subgroups from the acting group›

lemma (in group_action) stabilizer_subset:
"stabilizer G φ x ⊆ carrier G"
by (metis (no_types, lifting) mem_Collect_eq stabilizer_def subsetI)

lemma (in group_action) stabilizer_m_closed:
assumes "x ∈ E" "g1 ∈ (stabilizer G φ x)" "g2 ∈ (stabilizer G φ x)"
shows "(g1 ⊗ g2) ∈ (stabilizer G φ x)"
proof -
interpret group G
using group_hom group_hom.axioms(1) by auto

have "φ g1 x = x"
using assms stabilizer_def by fastforce
moreover have "φ g2 x = x"
using assms stabilizer_def by fastforce
moreover have g1: "g1 ∈ carrier G"
by (meson assms contra_subsetD stabilizer_subset)
moreover have g2: "g2 ∈ carrier G"
by (meson assms contra_subsetD stabilizer_subset)
ultimately have "φ (g1 ⊗ g2) x = x"
using composition_rule assms by simp

thus ?thesis
by (simp add: g1 g2 stabilizer_def)
qed

lemma (in group_action) stabilizer_one_closed:
assumes "x ∈ E"
shows "𝟭 ∈ (stabilizer G φ x)"
proof -
have "φ 𝟭 x = x"
by (metis assms id_eq_one restrict_apply')
thus ?thesis
using group_def group_hom group_hom.axioms(1) stabilizer_def by fastforce
qed

lemma (in group_action) stabilizer_m_inv_closed:
assumes "x ∈ E" "g ∈ (stabilizer G φ x)"
shows "(inv g) ∈ (stabilizer G φ x)"
proof -
interpret group G
using group_hom group_hom.axioms(1) by auto

have "φ g x = x"
using assms(2) stabilizer_def by fastforce
moreover have g: "g ∈ carrier G"
using assms(2) stabilizer_subset by blast
moreover have inv_g: "inv g ∈ carrier G"
ultimately have "φ (inv g) x = x"
using assms(1) orbit_sym_aux by blast

thus ?thesis by (simp add: inv_g stabilizer_def)
qed

theorem (in group_action) stabilizer_subgroup:
assumes "x ∈ E"
shows "subgroup (stabilizer G φ x) G"
unfolding subgroup_def
using stabilizer_subset stabilizer_m_closed stabilizer_one_closed
stabilizer_m_inv_closed assms by simp

subsection ‹The Orbit-Stabilizer Theorem›

text ‹In this subsection, we prove the Orbit-Stabilizer theorem.
Our approach is to show the existence of a bijection between
"rcosets (stabilizer G phi x)" and "orbit G phi x". Then we use
Lagrange's theorem to find the cardinal of the first set.›

subsubsection ‹Rcosets - Supporting Lemmas›

corollary (in group_action) stab_rcosets_not_empty:
assumes "x ∈ E" "R ∈ rcosets (stabilizer G φ x)"
shows "R ≠ {}"
using subgroup.rcosets_non_empty[OF stabilizer_subgroup[OF assms(1)] assms(2)] by simp

corollary (in group_action) diff_stabilizes:
assumes "x ∈ E" "R ∈ rcosets (stabilizer G φ x)"
shows "⋀g1 g2. ⟦ g1 ∈ R; g2 ∈ R ⟧ ⟹ g1 ⊗ (inv g2) ∈ stabilizer G φ x"
using group.diff_neutralizes[of G "stabilizer G φ x" R] stabilizer_subgroup[OF assms(1)]
assms(2) group_hom group_hom.axioms(1) by blast

subsubsection ‹Bijection Between Rcosets and an Orbit - Definition and Supporting Lemmas›

(* This definition could be easier if lcosets were available, and it's indeed a considerable
modification at Coset theory, since we could derive it easily from the definition of rcosets
following the same idea we use here: f: rcosets → lcosets, s.t. f R = (λg. inv g) ` R
is a bijection. *)

definition
orb_stab_fun :: "[_, ('a ⇒ 'b ⇒ 'b), 'a set, 'b] ⇒ 'b"
where "orb_stab_fun G φ R x = (φ (inv⇘G⇙ (SOME h. h ∈ R))) x"

lemma (in group_action) orbit_stab_fun_is_well_defined0:
assumes "x ∈ E" "R ∈ rcosets (stabilizer G φ x)"
shows "⋀g1 g2. ⟦ g1 ∈ R; g2 ∈ R ⟧ ⟹ (φ (inv g1)) x = (φ (inv g2)) x"
proof -
fix g1 g2 assume g1: "g1 ∈ R" and g2: "g2 ∈ R"
have R_carr: "R ⊆ carrier G"
using subgroup.rcosets_carrier[OF stabilizer_subgroup[OF assms(1)]]
assms(2) group_hom group_hom.axioms(1) by auto
from R_carr have g1_carr: "g1 ∈ carrier G" using g1 by blast
from R_carr have g2_carr: "g2 ∈ carrier G" using g2 by blast

have "g1 ⊗ (inv g2) ∈ stabilizer G φ x"
using diff_stabilizes[of x R g1 g2] assms g1 g2 by blast
hence "φ (g1 ⊗ (inv g2)) x = x"
hence "(φ (inv g1)) x = (φ (inv g1)) (φ (g1 ⊗ (inv g2)) x)" by simp
also have " ... = φ ((inv g1) ⊗ (g1 ⊗ (inv g2))) x"
using group_def assms(1) composition_rule g1_carr g2_carr
group_hom group_hom.axioms(1) monoid.m_closed by fastforce
also have " ... = φ (((inv g1) ⊗ g1) ⊗ (inv g2)) x"
using group_def g1_carr g2_carr group_hom group_hom.axioms(1) monoid.m_assoc by fastforce
finally show "(φ (inv g1)) x = (φ (inv g2)) x"
using group_def g1_carr g2_carr group.l_inv group_hom group_hom.axioms(1) by fastforce
qed

lemma (in group_action) orbit_stab_fun_is_well_defined1:
assumes "x ∈ E" "R ∈ rcosets (stabilizer G φ x)"
shows "⋀g. g ∈ R ⟹ (φ (inv (SOME h. h ∈ R))) x = (φ (inv g)) x"
by (meson assms orbit_stab_fun_is_well_defined0 someI_ex)

lemma (in group_action) orbit_stab_fun_is_inj:
assumes "x ∈ E"
and "R1 ∈ rcosets (stabilizer G φ x)"
and "R2 ∈ rcosets (stabilizer G φ x)"
and "φ (inv (SOME h. h ∈ R1)) x = φ (inv (SOME h. h ∈ R2)) x"
shows "R1 = R2"
proof -
have "(∃g1. g1 ∈ R1) ∧ (∃g2. g2 ∈ R2)"
using assms(1-3) stab_rcosets_not_empty by auto
then obtain g1 g2 where g1: "g1 ∈ R1" and g2: "g2 ∈ R2" by blast
hence g12_carr: "g1 ∈ carrier G ∧ g2 ∈ carrier G"
using subgroup.rcosets_carrier assms(1-3) group_hom
group_hom.axioms(1) stabilizer_subgroup by blast

then obtain r1 r2 where r1: "r1 ∈ carrier G" "R1 = (stabilizer G φ x) #> r1"
and r2: "r2 ∈ carrier G" "R2 = (stabilizer G φ x) #> r2"
using assms(1-3) unfolding RCOSETS_def by blast
then obtain s1 s2 where s1: "s1 ∈ (stabilizer G φ x)" "g1 = s1 ⊗ r1"
and s2: "s2 ∈ (stabilizer G φ x)" "g2 = s2 ⊗ r2"
using g1 g2 unfolding r_coset_def by blast

have "φ (inv g1) x = φ (inv (SOME h. h ∈ R1)) x"
using orbit_stab_fun_is_well_defined1[OF assms(1) assms(2) g1] by simp
also have " ... = φ (inv (SOME h. h ∈ R2)) x"
using assms(4) by simp
finally have "φ (inv g1) x = φ (inv g2) x"
using orbit_stab_fun_is_well_defined1[OF assms(1) assms(3) g2] by simp

hence "φ g2 (φ (inv g1) x) = φ g2 (φ (inv g2) x)" by simp
also have " ... = φ (g2 ⊗ (inv g2)) x"
using assms(1) composition_rule g12_carr group_hom group_hom.axioms(1) by fastforce
finally have "φ g2 (φ (inv g1) x) = x"
using g12_carr assms(1) group.r_inv group_hom group_hom.axioms(1)
id_eq_one restrict_apply by metis
hence "φ (g2 ⊗ (inv g1)) x = x"
using assms(1) composition_rule g12_carr group_hom group_hom.axioms(1) by fastforce
hence "g2 ⊗ (inv g1) ∈ (stabilizer G φ x)"
using g12_carr group.subgroup_self group_hom group_hom.axioms(1)
mem_Collect_eq stabilizer_def subgroup_def by (metis (mono_tags, lifting))
then obtain s where s: "s ∈ (stabilizer G φ x)" "s = g2 ⊗ (inv g1)" by blast

let ?h = "s ⊗ g1"
have "?h = s ⊗ (s1 ⊗ r1)" by (simp add: s1)
hence "?h = (s ⊗ s1) ⊗ r1"
using stabilizer_subgroup[OF assms(1)] group_def group_hom
group_hom.axioms(1) monoid.m_assoc r1 s s1 subgroup.mem_carrier by fastforce
hence inR1: "?h ∈ (stabilizer G φ x) #> r1" unfolding r_coset_def
using stabilizer_subgroup[OF assms(1)] assms(1) s s1 stabilizer_m_closed by auto

have "?h = g2" using s stabilizer_subgroup[OF assms(1)] g12_carr group.inv_solve_right
group_hom group_hom.axioms(1) subgroup.mem_carrier by metis
hence inR2: "?h ∈ (stabilizer G φ x) #> r2"
using g2 r2 by blast

have "R1 ∩ R2 ≠ {}" using inR1 inR2 r1 r2 by blast
thus ?thesis
using stabilizer_subgroup group.rcos_disjoint[of G "stabilizer G φ x"] assms group_hom group_hom.axioms(1)
unfolding disjnt_def pairwise_def  by blast
qed

lemma (in group_action) orbit_stab_fun_is_surj:
assumes "x ∈ E" "y ∈ orbit G φ x"
shows "∃R ∈ rcosets (stabilizer G φ x). φ (inv (SOME h. h ∈ R)) x = y"
proof -
have "∃g ∈ carrier G. (φ g) x = y"
using assms(2) unfolding orbit_def by blast
then obtain g where g: "g ∈ carrier G ∧ (φ g) x = y" by blast

let ?R = "(stabilizer G φ x) #> (inv g)"
have R: "?R ∈ rcosets (stabilizer G φ x)"
unfolding RCOSETS_def using g group_hom group_hom.axioms(1) by fastforce
moreover have "𝟭 ⊗ (inv g) ∈ ?R"
unfolding r_coset_def using assms(1) stabilizer_one_closed by auto
ultimately have "φ (inv (SOME h. h ∈ ?R)) x = φ (inv (𝟭 ⊗ (inv g))) x"
using orbit_stab_fun_is_well_defined1[OF assms(1)] by simp
also have " ... = (φ g) x"
using group_def g group_hom group_hom.axioms(1) monoid.l_one by fastforce
finally have "φ (inv (SOME h. h ∈ ?R)) x = y"
using g by simp
thus ?thesis using R by blast
qed

proposition (in group_action) orbit_stab_fun_is_bij:
assumes "x ∈ E"
shows "bij_betw (λR. (φ (inv (SOME h. h ∈ R))) x) (rcosets (stabilizer G φ x)) (orbit G φ x)"
unfolding bij_betw_def
proof
show "inj_on (λR. φ (inv (SOME h. h ∈ R)) x) (rcosets stabilizer G φ x)"
using orbit_stab_fun_is_inj[OF assms(1)] by (simp add: inj_on_def)
next
have "⋀R. R ∈ (rcosets stabilizer G φ x) ⟹ φ (inv (SOME h. h ∈ R)) x ∈ orbit G φ x "
proof -
fix R assume R: "R ∈ (rcosets stabilizer G φ x)"
then obtain g where g: "g ∈ R"
using assms stab_rcosets_not_empty by auto
hence "φ (inv (SOME h. h ∈ R)) x = φ (inv g) x"
using R  assms orbit_stab_fun_is_well_defined1 by blast
thus "φ (inv (SOME h. h ∈ R)) x ∈ orbit G φ x" unfolding orbit_def
using subgroup.rcosets_carrier group_hom group_hom.axioms(1)
g R assms stabilizer_subgroup by fastforce
qed
moreover have "orbit G φ x ⊆ (λR. φ (inv (SOME h. h ∈ R)) x) ` (rcosets stabilizer G φ x)"
using assms orbit_stab_fun_is_surj by fastforce
ultimately show "(λR. φ (inv (SOME h. h ∈ R)) x) ` (rcosets stabilizer G φ x) = orbit G φ x "
using assms set_eq_subset by blast
qed

subsubsection ‹The Theorem›

theorem (in group_action) orbit_stabilizer_theorem:
assumes "x ∈ E"
shows "card (orbit G φ x) * card (stabilizer G φ x) = order G"
proof -
have "card (rcosets stabilizer G φ x) = card (orbit G φ x)"
using orbit_stab_fun_is_bij[OF assms(1)] bij_betw_same_card by blast
moreover have "card (rcosets stabilizer G φ x) * card (stabilizer G φ x) = order G"
using stabilizer_subgroup assms group.lagrange group_hom group_hom.axioms(1) by blast
ultimately show ?thesis by auto
qed

subsection ‹The Burnside's Lemma›

subsubsection ‹Sums and Cardinals›

lemma card_as_sums:
assumes "A = {x ∈ B. P x}" "finite B"
shows "card A = (∑x∈B. (if P x then 1 else 0))"
proof -
have "A ⊆ B" using assms(1) by blast
have "card A = (∑x∈A. 1)" by simp
also have " ... = (∑x∈A. (if P x then 1 else 0))"
also have " ... = (∑x∈A. (if P x then 1 else 0)) + (∑x∈(B - A). (if P x then 1 else 0))"
using assms(1) by auto
finally show "card A = (∑x∈B. (if P x then 1 else 0))"
using ‹A ⊆ B› add.commute assms(2) sum.subset_diff by metis
qed

lemma sum_invertion:
"⟦ finite A; finite B ⟧ ⟹ (∑x∈A. ∑y∈B. f x y) = (∑y∈B. ∑x∈A. f x y)"
proof (induct set: finite)
case empty thus ?case by simp
next
case (insert x A')
have "(∑x∈insert x A'. ∑y∈B. f x y) = (∑y∈B. f x y) + (∑x∈A'. ∑y∈B. f x y)"
also have " ... = (∑y∈B. f x y) + (∑y∈B. ∑x∈A'. f x y)"
using insert.hyps by (simp add: insert.prems)
also have " ... = (∑y∈B. (f x y) + (∑x∈A'. f x y))"
finally have "(∑x∈insert x A'. ∑y∈B. f x y) = (∑y∈B. ∑x∈insert x A'. f x y)"
using sum.swap by blast
thus ?case by simp
qed

lemma (in group_action) card_stablizer_sum:
assumes "finite (carrier G)" "orb ∈ (orbits G E φ)"
shows "(∑x ∈ orb. card (stabilizer G φ x)) = order G"
proof -
obtain x where x:"x ∈ E" and orb:"orb = orbit G φ x"
using assms(2) unfolding orbits_def by blast
have "⋀y. y ∈ orb ⟹ card (stabilizer G φ x) = card (stabilizer G φ y)"
proof -
fix y assume "y ∈ orb"
hence y: "y ∈ E ∧ y ∈ orbit G φ x"
using x orb assms(2) orbits_coverture by auto
hence same_orbit: "(orbit G φ x) = (orbit G φ y)"
using disjoint_union[of "orbit G φ x" "orbit G φ y"] orbit_refl x
unfolding orbits_def by auto
have "card (orbit G φ x) * card (stabilizer G φ x) =
card (orbit G φ y) * card (stabilizer G φ y)"
using y assms(1) x orbit_stabilizer_theorem by simp
hence "card (orbit G φ x) * card (stabilizer G φ x) =
card (orbit G φ x) * card (stabilizer G φ y)" using same_orbit by simp
moreover have "orbit G φ x ≠ {} ∧ finite (orbit G φ x)"
using y orbit_def[of G φ x] assms(1) by auto
hence "card (orbit G φ x) > 0"
ultimately show "card (stabilizer G φ x) = card (stabilizer G φ y)" by auto
qed
hence "(∑x ∈ orb. card (stabilizer G φ x)) = (∑y ∈ orb. card (stabilizer G φ x))" by auto
also have " ... = card (stabilizer G φ x) * (∑y ∈ orb. 1)" by simp
also have " ... = card (stabilizer G φ x) * card (orbit G φ x)"
using orb by auto
finally show "(∑x ∈ orb. card (stabilizer G φ x)) = order G"
by (metis mult.commute orbit_stabilizer_theorem x)
qed

subsubsection ‹The Lemma›

theorem (in group_action) burnside:
assumes "finite (carrier G)" "finite E"
shows "card (orbits G E φ) * order G = (∑g ∈ carrier G. card(invariants E φ g))"
proof -
have "(∑g ∈ carrier G. card(invariants E φ g)) =
(∑g ∈ carrier G. ∑x ∈ E. (if (φ g) x = x then 1 else 0))"
by (simp add: assms(2) card_as_sums invariants_def)
also have " ... = (∑x ∈ E. ∑g ∈ carrier G. (if (φ g) x = x then 1 else 0))"
using sum_invertion[where ?f = "λ g x. (if (φ g) x = x then 1 else 0)"] assms by auto
also have " ... = (∑x ∈ E. card (stabilizer G φ x))"
by (simp add: assms(1) card_as_sums stabilizer_def)
also have " ... = (∑orbit ∈ (orbits G E φ). ∑x ∈ orbit. card (stabilizer G φ x))"
using disjoint_sum orbits_coverture assms(2) by metis
also have " ... = (∑orbit ∈ (orbits G E φ). order G)"
finally have "(∑g ∈ carrier G. card(invariants E φ g)) = card (orbits G E φ) * order G" by simp
thus ?thesis by simp
qed

subsection ‹Action by Conjugation›

subsubsection ‹Action Over Itself›

text ‹A Group Acts by Conjugation Over Itself›

lemma (in group) conjugation_is_inj:
assumes "g ∈ carrier G" "h1 ∈ carrier G" "h2 ∈ carrier G"
and "g ⊗ h1 ⊗ (inv g) = g ⊗ h2 ⊗ (inv g)"
shows "h1 = h2"
using assms by auto

lemma (in group) conjugation_is_surj:
assumes "g ∈ carrier G" "h ∈ carrier G"
shows "g ⊗ ((inv g) ⊗ h ⊗ g) ⊗ (inv g) = h"
using assms m_assoc inv_closed inv_inv m_closed monoid_axioms r_inv r_one
by metis

lemma (in group) conjugation_is_bij:
assumes "g ∈ carrier G"
shows "bij_betw (λh ∈ carrier G. g ⊗ h ⊗ (inv g)) (carrier G) (carrier G)"
(is "bij_betw ?φ (carrier G) (carrier G)")
unfolding bij_betw_def
proof
show "inj_on ?φ (carrier G)"
using conjugation_is_inj by (simp add: assms inj_on_def)
next
have S: "⋀ h. h ∈ carrier G ⟹ (inv g) ⊗ h ⊗ g ∈ carrier G"
using assms by blast
have "⋀ h. h ∈ carrier G ⟹ ?φ ((inv g) ⊗ h ⊗ g) = h"
using assms by (simp add: conjugation_is_surj)
hence "carrier G ⊆ ?φ ` carrier G"
using S image_iff by fastforce
moreover have "⋀ h. h ∈ carrier G ⟹ ?φ h ∈ carrier G"
using assms by simp
hence "?φ ` carrier G ⊆ carrier G" by blast
ultimately show "?φ ` carrier G = carrier G" by blast
qed

lemma(in group) conjugation_is_hom:
"(λg. λh ∈ carrier G. g ⊗ h ⊗ inv g) ∈ hom G (BijGroup (carrier G))"
unfolding hom_def
proof -
let ?ψ = "λg. λh. g ⊗ h ⊗ inv g"
let ?φ = "λg. restrict (?ψ g) (carrier G)"

(* First, we prove that ?φ: G → Bij(G) is well defined *)
have Step0: "⋀ g. g ∈ carrier G ⟹ (?φ g) ∈ Bij (carrier G)"
using Bij_def conjugation_is_bij by fastforce
hence Step1: "?φ: carrier G → carrier (BijGroup (carrier G))"
unfolding BijGroup_def by simp

(* Second, we prove that ?φ is a homomorphism *)
have "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
(⋀ h. h ∈ carrier G ⟹ ?ψ (g1 ⊗ g2) h = (?φ g1) ((?φ g2) h))"
proof -
fix g1 g2 h assume g1: "g1 ∈ carrier G" and g2: "g2 ∈ carrier G" and h: "h ∈ carrier G"
have "inv (g1 ⊗ g2) = (inv g2) ⊗ (inv g1)"
using g1 g2 by (simp add: inv_mult_group)
thus "?ψ (g1 ⊗ g2) h  = (?φ g1) ((?φ g2) h)"
by (simp add: g1 g2 h m_assoc)
qed
hence "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
(λ h ∈ carrier G. ?ψ (g1 ⊗ g2) h) = (λ h ∈ carrier G. (?φ g1) ((?φ g2) h))" by auto
hence Step2: "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
?φ (g1 ⊗ g2) = (?φ g1) ⊗⇘BijGroup (carrier G)⇙ (?φ g2)"
unfolding BijGroup_def by (simp add: Step0 compose_def)

(* Finally, we combine both results to prove the lemma *)
thus "?φ ∈ {h: carrier G → carrier (BijGroup (carrier G)).
(∀x ∈ carrier G. ∀y ∈ carrier G. h (x ⊗ y) = h x ⊗⇘BijGroup (carrier G)⇙ h y)}"
using Step1 Step2 by auto
qed

theorem (in group) action_by_conjugation:
"group_action G (carrier G) (λg. (λh ∈ carrier G. g ⊗ h ⊗ (inv g)))"
unfolding group_action_def group_hom_def using conjugation_is_hom
by (simp add: group_BijGroup group_hom_axioms.intro is_group)

subsubsection ‹Action Over The Set of Subgroups›

text ‹A Group Acts by Conjugation Over The Set of Subgroups›

lemma (in group) subgroup_conjugation_is_inj_aux:
assumes "g ∈ carrier G" "H1 ⊆ carrier G" "H2 ⊆ carrier G"
and "g <# H1 #> (inv g) = g <# H2 #> (inv g)"
shows "H1 ⊆ H2"
proof
fix h1 assume h1: "h1 ∈ H1"
hence "g ⊗ h1 ⊗ (inv g) ∈ g <# H1 #> (inv g)"
unfolding l_coset_def r_coset_def using assms by blast
hence "g ⊗ h1 ⊗ (inv g) ∈ g <# H2 #> (inv g)"
using assms by auto
hence "∃h2 ∈ H2. g ⊗ h1 ⊗ (inv g) = g ⊗ h2 ⊗ (inv g)"
unfolding l_coset_def r_coset_def by blast
then obtain h2 where "h2 ∈ H2 ∧ g ⊗ h1 ⊗ (inv g) = g ⊗ h2 ⊗ (inv g)" by blast
thus "h1 ∈ H2"
using assms conjugation_is_inj h1 by blast
qed

lemma (in group) subgroup_conjugation_is_inj:
assumes "g ∈ carrier G" "H1 ⊆ carrier G" "H2 ⊆ carrier G"
and "g <# H1 #> (inv g) = g <# H2 #> (inv g)"
shows "H1 = H2"
using subgroup_conjugation_is_inj_aux assms set_eq_subset by metis

lemma (in group) subgroup_conjugation_is_surj0:
assumes "g ∈ carrier G" "H ⊆ carrier G"
shows "g <# ((inv g) <# H #> g) #> (inv g) = H"
using coset_assoc assms coset_mult_assoc l_coset_subset_G lcos_m_assoc

lemma (in group) subgroup_conjugation_is_surj1:
assumes "g ∈ carrier G" "subgroup H G"
shows "subgroup ((inv g) <# H #> g) G"
proof
show "𝟭 ∈ inv g <# H #> g"
proof -
have "𝟭 ∈ H" by (simp add: assms(2) subgroup.one_closed)
hence "inv g ⊗ 𝟭 ⊗ g ∈ inv g <# H #> g"
unfolding l_coset_def r_coset_def by blast
thus "𝟭 ∈ inv g <# H #> g" using assms by simp
qed
next
show "inv g <# H #> g ⊆ carrier G"
proof
fix x assume "x ∈ inv g <# H #> g"
hence "∃h ∈ H. x = (inv g) ⊗ h ⊗ g"
unfolding r_coset_def l_coset_def by blast
hence "∃h ∈ (carrier G). x = (inv g) ⊗ h ⊗ g"
by (meson assms subgroup.mem_carrier)
thus "x ∈ carrier G" using assms by blast
qed
next
show " ⋀ x y. ⟦ x ∈ inv g <# H #> g; y ∈ inv g <# H #> g ⟧ ⟹ x ⊗ y ∈ inv g <# H #> g"
proof -
fix x y assume "x ∈ inv g <# H #> g"  "y ∈ inv g <# H #> g"
then obtain h1 h2 where h12: "h1 ∈ H" "h2 ∈ H" and "x = (inv g) ⊗ h1 ⊗ g ∧ y = (inv g) ⊗ h2 ⊗ g"
unfolding l_coset_def r_coset_def by blast
hence "x ⊗ y = ((inv g) ⊗ h1 ⊗ g) ⊗ ((inv g) ⊗ h2 ⊗ g)" by blast
also have "… = ((inv g) ⊗ h1 ⊗ (g ⊗ inv g) ⊗ h2 ⊗ g)"
using h12 assms inv_closed  m_assoc m_closed subgroup.mem_carrier [OF ‹subgroup H G›] by presburger
also have "… = ((inv g) ⊗ (h1 ⊗ h2) ⊗ g)"
by (simp add: h12 assms m_assoc subgroup.mem_carrier [OF ‹subgroup H G›])
finally have "∃ h ∈ H. x ⊗ y = (inv g) ⊗ h ⊗ g"
by (meson assms(2) h12 subgroup_def)
thus "x ⊗ y ∈ inv g <# H #> g"
unfolding l_coset_def r_coset_def by blast
qed
next
show "⋀x. x ∈ inv g <# H #> g ⟹ inv x ∈ inv g <# H #> g"
proof -
fix x assume "x ∈ inv g <# H #> g"
hence "∃h ∈ H. x = (inv g) ⊗ h ⊗ g"
unfolding r_coset_def l_coset_def by blast
then obtain h where h: "h ∈ H ∧ x = (inv g) ⊗ h ⊗ g" by blast
hence "x ⊗ (inv g) ⊗ (inv h) ⊗ g = 𝟭"
using assms m_assoc monoid_axioms by (simp add: subgroup.mem_carrier)
hence "inv x = (inv g) ⊗ (inv h) ⊗ g"
using assms h inv_mult_group m_assoc monoid_axioms by (simp add: subgroup.mem_carrier)
moreover have "inv h ∈ H"
by (simp add: assms h subgroup.m_inv_closed)
ultimately show "inv x ∈ inv g <# H #> g" unfolding r_coset_def l_coset_def by blast
qed
qed

lemma (in group) subgroup_conjugation_is_surj2:
assumes "g ∈ carrier G" "subgroup H G"
shows "subgroup (g <# H #> (inv g)) G"
using subgroup_conjugation_is_surj1 by (metis assms inv_closed inv_inv)

lemma (in group) subgroup_conjugation_is_bij:
assumes "g ∈ carrier G"
shows "bij_betw (λH ∈ {H. subgroup H G}. g <# H #> (inv g)) {H. subgroup H G} {H. subgroup H G}"
(is "bij_betw ?φ {H. subgroup H G} {H. subgroup H G}")
unfolding bij_betw_def
proof
show "inj_on ?φ {H. subgroup H G}"
using subgroup_conjugation_is_inj assms inj_on_def subgroup.subset
by (metis (mono_tags, lifting) inj_on_restrict_eq mem_Collect_eq)
next
have "⋀H. H ∈ {H. subgroup H G} ⟹ ?φ ((inv g) <# H #> g) = H"
by (simp add: assms subgroup.subset subgroup_conjugation_is_surj0
subgroup_conjugation_is_surj1 is_group)
hence "⋀H. H ∈ {H. subgroup H G} ⟹ ∃H' ∈ {H. subgroup H G}. ?φ H' = H"
using assms subgroup_conjugation_is_surj1 by fastforce
thus "?φ ` {H. subgroup H G} = {H. subgroup H G}"
using subgroup_conjugation_is_surj2 assms by auto
qed

lemma (in group) subgroup_conjugation_is_hom:
"(λg. λH ∈ {H. subgroup H G}. g <# H #> (inv g)) ∈ hom G (BijGroup {H. subgroup H G})"
unfolding hom_def
proof -
(* We follow the exact same structure of conjugation_is_hom's proof *)
let ?ψ = "λg. λH. g <# H #> (inv g)"
let ?φ = "λg. restrict (?ψ g) {H. subgroup H G}"

have Step0: "⋀ g. g ∈ carrier G ⟹ (?φ g) ∈ Bij {H. subgroup H G}"
using Bij_def subgroup_conjugation_is_bij by fastforce
hence Step1: "?φ: carrier G → carrier (BijGroup {H. subgroup H G})"
unfolding BijGroup_def by simp

have "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
(⋀ H. H ∈ {H. subgroup H G} ⟹ ?ψ (g1 ⊗ g2) H = (?φ g1) ((?φ g2) H))"
proof -
fix g1 g2 H assume g1: "g1 ∈ carrier G" and g2: "g2 ∈ carrier G" and H': "H ∈ {H. subgroup H G}"
hence H: "subgroup H G" by simp
have "(?φ g1) ((?φ g2) H) = g1 <# (g2 <# H #> (inv g2)) #> (inv g1)"
by (simp add: H g2 subgroup_conjugation_is_surj2)
also have " ... = g1 <# (g2 <# H) #> ((inv g2) ⊗ (inv g1))"
by (simp add: H coset_mult_assoc g1 g2 group.coset_assoc
is_group l_coset_subset_G subgroup.subset)
also have " ... = g1 <# (g2 <# H) #> inv (g1 ⊗ g2)"
using g1 g2 by (simp add: inv_mult_group)
finally have "(?φ g1) ((?φ g2) H) = ?ψ (g1 ⊗ g2) H"
by (simp add: H g1 g2 lcos_m_assoc subgroup.subset)
thus "?ψ (g1 ⊗ g2) H = (?φ g1) ((?φ g2) H)" by auto
qed
hence "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
(λH ∈ {H. subgroup H G}. ?ψ (g1 ⊗ g2) H) = (λH ∈ {H. subgroup H G}. (?φ g1) ((?φ g2) H))"
by (meson restrict_ext)
hence Step2: "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
?φ (g1 ⊗ g2) = (?φ g1) ⊗⇘BijGroup {H. subgroup H G}⇙ (?φ g2)"
unfolding BijGroup_def by (simp add: Step0 compose_def)

show "?φ ∈ {h: carrier G → carrier (BijGroup {H. subgroup H G}).
∀x∈carrier G. ∀y∈carrier G. h (x ⊗ y) = h x ⊗⇘BijGroup {H. subgroup H G}⇙ h y}"
using Step1 Step2 by auto
qed

theorem (in group) action_by_conjugation_on_subgroups_set:
"group_action G {H. subgroup H G} (λg. λH ∈ {H. subgroup H G}. g <# H #> (inv g))"
unfolding group_action_def group_hom_def using subgroup_conjugation_is_hom
by (simp add: group_BijGroup group_hom_axioms.intro is_group)

subsubsection ‹Action Over The Power Set›

text ‹A Group Acts by Conjugation Over The Power Set›

lemma (in group) subset_conjugation_is_bij:
assumes "g ∈ carrier G"
shows "bij_betw (λH ∈ {H. H ⊆ carrier G}. g <# H #> (inv g)) {H. H ⊆ carrier G} {H. H ⊆ carrier G}"
(is "bij_betw ?φ {H. H ⊆ carrier G} {H. H ⊆ carrier G}")
unfolding bij_betw_def
proof
show "inj_on ?φ {H. H ⊆ carrier G}"
using subgroup_conjugation_is_inj assms inj_on_def
by (metis (mono_tags, lifting) inj_on_restrict_eq mem_Collect_eq)
next
have "⋀H. H ∈ {H. H ⊆ carrier G} ⟹ ?φ ((inv g) <# H #> g) = H"
by (simp add: assms l_coset_subset_G r_coset_subset_G subgroup_conjugation_is_surj0)
hence "⋀H. H ∈ {H. H ⊆ carrier G} ⟹ ∃H' ∈ {H. H ⊆ carrier G}. ?φ H' = H"
by (metis assms l_coset_subset_G mem_Collect_eq r_coset_subset_G subgroup_def subgroup_self)
hence "{H. H ⊆ carrier G} ⊆ ?φ ` {H. H ⊆ carrier G}" by blast
moreover have "?φ ` {H. H ⊆ carrier G} ⊆ {H. H ⊆ carrier G}"
by clarsimp (meson assms contra_subsetD inv_closed l_coset_subset_G r_coset_subset_G)
ultimately show "?φ ` {H. H ⊆ carrier G} = {H. H ⊆ carrier G}" by simp
qed

lemma (in group) subset_conjugation_is_hom:
"(λg. λH ∈ {H. H ⊆ carrier G}. g <# H #> (inv g)) ∈ hom G (BijGroup {H. H ⊆ carrier G})"
unfolding hom_def
proof -
(* We follow the exact same structure of conjugation_is_hom's proof *)
let ?ψ = "λg. λH. g <# H #> (inv g)"
let ?φ = "λg. restrict (?ψ g) {H. H ⊆ carrier G}"

have Step0: "⋀ g. g ∈ carrier G ⟹ (?φ g) ∈ Bij {H. H ⊆ carrier G}"
using Bij_def subset_conjugation_is_bij by fastforce
hence Step1: "?φ: carrier G → carrier (BijGroup {H. H ⊆ carrier G})"
unfolding BijGroup_def by simp

have "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
(⋀ H. H ∈ {H. H ⊆ carrier G} ⟹ ?ψ (g1 ⊗ g2) H = (?φ g1) ((?φ g2) H))"
proof -
fix g1 g2 H assume g1: "g1 ∈ carrier G" and g2: "g2 ∈ carrier G" and H: "H ∈ {H. H ⊆ carrier G}"
hence "(?φ g1) ((?φ g2) H) = g1 <# (g2 <# H #> (inv g2)) #> (inv g1)"
using l_coset_subset_G r_coset_subset_G by auto
also have " ... = g1 <# (g2 <# H) #> ((inv g2) ⊗ (inv g1))"
using H coset_assoc coset_mult_assoc g1 g2 l_coset_subset_G by auto
also have " ... = g1 <# (g2 <# H) #> inv (g1 ⊗ g2)"
using g1 g2 by (simp add: inv_mult_group)
finally have "(?φ g1) ((?φ g2) H) = ?ψ (g1 ⊗ g2) H"
using H g1 g2 lcos_m_assoc by force
thus "?ψ (g1 ⊗ g2) H = (?φ g1) ((?φ g2) H)" by auto
qed
hence "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
(λH ∈ {H. H ⊆ carrier G}. ?ψ (g1 ⊗ g2) H) = (λH ∈ {H. H ⊆ carrier G}. (?φ g1) ((?φ g2) H))"
by (meson restrict_ext)
hence Step2: "⋀ g1 g2. ⟦ g1 ∈ carrier G; g2 ∈ carrier G ⟧ ⟹
?φ (g1 ⊗ g2) = (?φ g1) ⊗⇘BijGroup {H. H ⊆ carrier G}⇙ (?φ g2)"
unfolding BijGroup_def by (simp add: Step0 compose_def)

show "?φ ∈ {h: carrier G → carrier (BijGroup {H. H ⊆ carrier G}).
∀x∈carrier G. ∀y∈carrier G. h (x ⊗ y) = h x ⊗⇘BijGroup {H. H ⊆ carrier G}⇙ h y}"
using Step1 Step2 by auto
qed

theorem (in group) action_by_conjugation_on_power_set:
"group_action G {H. H ⊆ carrier G} (λg. λH ∈ {H. H ⊆ carrier G}. g <# H #> (inv g))"
unfolding group_action_def group_hom_def using subset_conjugation_is_hom
by (simp add: group_BijGroup group_hom_axioms.intro is_group)

corollary (in group) normalizer_imp_subgroup:
assumes "H ⊆ carrier G"
shows "subgroup (normalizer G H) G"
unfolding normalizer_def
using group_action.stabilizer_subgroup[OF action_by_conjugation_on_power_set] assms by auto

subsection ‹Subgroup of an Acting Group›

text ‹A Subgroup of an Acting Group Induces an Action›

lemma (in group_action) induced_homomorphism:
assumes "subgroup H G"
shows "φ ∈ hom (G ⦇carrier := H⦈) (BijGroup E)"
unfolding hom_def apply simp
proof -
have S0: "H ⊆ carrier G" by (meson assms subgroup_def)
hence "φ: H → carrier (BijGroup E)"
by (simp add: BijGroup_def bij_prop0 subset_eq)
thus "φ: H → carrier (BijGroup E) ∧ (∀x ∈ H. ∀y ∈ H. φ (x ⊗ y) = φ x ⊗⇘BijGroup E⇙ φ y)"
by (simp add: S0  group_hom group_hom.hom_mult rev_subsetD)
qed

theorem (in group_action) induced_action:
assumes "subgroup H G"
shows "group_action (G ⦇carrier := H⦈) E φ"
unfolding group_action_def group_hom_def
using induced_homomorphism assms group.subgroup_imp_group group_BijGroup
group_hom group_hom.axioms(1) group_hom_axioms_def by blast

end
```