(* Title: HOL/Algebra/Coset.thy ID: $Id: Coset.thy,v 1.16 2005/06/17 14:13:05 haftmann Exp $ Author: Florian Kammueller, with new proofs by L C Paulson *) header{*Cosets and Quotient Groups*} theory Coset imports Group Exponent begin constdefs (structure G) r_coset :: "[_, 'a set, 'a] => 'a set" (infixl "#>\<index>" 60) "H #> a ≡ \<Union>h∈H. {h ⊗ a}" l_coset :: "[_, 'a, 'a set] => 'a set" (infixl "<#\<index>" 60) "a <# H ≡ \<Union>h∈H. {a ⊗ h}" RCOSETS :: "[_, 'a set] => ('a set)set" ("rcosets\<index> _" [81] 80) "rcosets H ≡ \<Union>a∈carrier G. {H #> a}" set_mult :: "[_, 'a set ,'a set] => 'a set" (infixl "<#>\<index>" 60) "H <#> K ≡ \<Union>h∈H. \<Union>k∈K. {h ⊗ k}" SET_INV :: "[_,'a set] => 'a set" ("set'_inv\<index> _" [81] 80) "set_inv H ≡ \<Union>h∈H. {inv h}" locale normal = subgroup + group + assumes coset_eq: "(∀x ∈ carrier G. H #> x = x <# H)" syntax "@normal" :: "['a set, ('a, 'b) monoid_scheme] => bool" (infixl "\<lhd>" 60) translations "H \<lhd> G" == "normal H G" subsection {*Basic Properties of Cosets*} lemma (in group) coset_mult_assoc: "[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> (M #> g) #> h = M #> (g ⊗ h)" by (force simp add: r_coset_def m_assoc) lemma (in group) coset_mult_one [simp]: "M ⊆ carrier G ==> M #> \<one> = M" by (force simp add: r_coset_def) lemma (in group) coset_mult_inv1: "[| M #> (x ⊗ (inv y)) = M; x ∈ carrier G ; y ∈ carrier G; M ⊆ carrier G |] ==> M #> x = M #> y" apply (erule subst [of concl: "%z. M #> x = z #> y"]) apply (simp add: coset_mult_assoc m_assoc) done lemma (in group) coset_mult_inv2: "[| M #> x = M #> y; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |] ==> M #> (x ⊗ (inv y)) = M " apply (simp add: coset_mult_assoc [symmetric]) apply (simp add: coset_mult_assoc) done lemma (in group) coset_join1: "[| H #> x = H; x ∈ carrier G; subgroup H G |] ==> x ∈ H" apply (erule subst) apply (simp add: r_coset_def) apply (blast intro: l_one subgroup.one_closed sym) done lemma (in group) solve_equation: "[|subgroup H G; x ∈ H; y ∈ H|] ==> ∃h∈H. y = h ⊗ x" apply (rule bexI [of _ "y ⊗ (inv x)"]) apply (auto simp add: subgroup.m_closed subgroup.m_inv_closed m_assoc subgroup.subset [THEN subsetD]) done lemma (in group) repr_independence: "[|y ∈ H #> x; x ∈ carrier G; subgroup H G|] ==> H #> x = H #> y" by (auto simp add: r_coset_def m_assoc [symmetric] subgroup.subset [THEN subsetD] subgroup.m_closed solve_equation) lemma (in group) coset_join2: "[|x ∈ carrier G; subgroup H G; x∈H|] ==> H #> x = H" --{*Alternative proof is to put @{term "x=\<one>"} in @{text repr_independence}.*} by (force simp add: subgroup.m_closed r_coset_def solve_equation) lemma (in group) r_coset_subset_G: "[| H ⊆ carrier G; x ∈ carrier G |] ==> H #> x ⊆ carrier G" by (auto simp add: r_coset_def) lemma (in group) rcosI: "[| h ∈ H; H ⊆ carrier G; x ∈ carrier G|] ==> h ⊗ x ∈ H #> x" by (auto simp add: r_coset_def) lemma (in group) rcosetsI: "[|H ⊆ carrier G; x ∈ carrier G|] ==> H #> x ∈ rcosets H" by (auto simp add: RCOSETS_def) text{*Really needed?*} lemma (in group) transpose_inv: "[| x ⊗ y = z; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (inv x) ⊗ z = y" by (force simp add: m_assoc [symmetric]) lemma (in group) rcos_self: "[| x ∈ carrier G; subgroup H G |] ==> x ∈ H #> x" apply (simp add: r_coset_def) apply (blast intro: sym l_one subgroup.subset [THEN subsetD] subgroup.one_closed) done subsection {* Normal subgroups *} lemma normal_imp_subgroup: "H \<lhd> G ==> subgroup H G" by (simp add: normal_def subgroup_def) lemma (in group) normalI: "subgroup H G ==> (∀x ∈ carrier G. H #> x = x <# H) ==> H \<lhd> G"; by (simp add: normal_def normal_axioms_def prems) lemma (in normal) inv_op_closed1: "[|x ∈ carrier G; h ∈ H|] ==> (inv x) ⊗ h ⊗ x ∈ H" apply (insert coset_eq) apply (auto simp add: l_coset_def r_coset_def) apply (drule bspec, assumption) apply (drule equalityD1 [THEN subsetD], blast, clarify) apply (simp add: m_assoc) apply (simp add: m_assoc [symmetric]) done lemma (in normal) inv_op_closed2: "[|x ∈ carrier G; h ∈ H|] ==> x ⊗ h ⊗ (inv x) ∈ H" apply (subgoal_tac "inv (inv x) ⊗ h ⊗ (inv x) ∈ H") apply (simp add: ); apply (blast intro: inv_op_closed1) done text{*Alternative characterization of normal subgroups*} lemma (in group) normal_inv_iff: "(N \<lhd> G) = (subgroup N G & (∀x ∈ carrier G. ∀h ∈ N. x ⊗ h ⊗ (inv x) ∈ N))" (is "_ = ?rhs") proof assume N: "N \<lhd> G" show ?rhs by (blast intro: N normal.inv_op_closed2 normal_imp_subgroup) next assume ?rhs hence sg: "subgroup N G" and closed: "!!x. x∈carrier G ==> ∀h∈N. x ⊗ h ⊗ inv x ∈ N" by auto hence sb: "N ⊆ carrier G" by (simp add: subgroup.subset) show "N \<lhd> G" proof (intro normalI [OF sg], simp add: l_coset_def r_coset_def, clarify) fix x assume x: "x ∈ carrier G" show "(\<Union>h∈N. {h ⊗ x}) = (\<Union>h∈N. {x ⊗ h})" proof show "(\<Union>h∈N. {h ⊗ x}) ⊆ (\<Union>h∈N. {x ⊗ h})" proof clarify fix n assume n: "n ∈ N" show "n ⊗ x ∈ (\<Union>h∈N. {x ⊗ h})" proof from closed [of "inv x"] show "inv x ⊗ n ⊗ x ∈ N" by (simp add: x n) show "n ⊗ x ∈ {x ⊗ (inv x ⊗ n ⊗ x)}" by (simp add: x n m_assoc [symmetric] sb [THEN subsetD]) qed qed next show "(\<Union>h∈N. {x ⊗ h}) ⊆ (\<Union>h∈N. {h ⊗ x})" proof clarify fix n assume n: "n ∈ N" show "x ⊗ n ∈ (\<Union>h∈N. {h ⊗ x})" proof show "x ⊗ n ⊗ inv x ∈ N" by (simp add: x n closed) show "x ⊗ n ∈ {x ⊗ n ⊗ inv x ⊗ x}" by (simp add: x n m_assoc sb [THEN subsetD]) qed qed qed qed qed subsection{*More Properties of Cosets*} lemma (in group) lcos_m_assoc: "[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> g <# (h <# M) = (g ⊗ h) <# M" by (force simp add: l_coset_def m_assoc) lemma (in group) lcos_mult_one: "M ⊆ carrier G ==> \<one> <# M = M" by (force simp add: l_coset_def) lemma (in group) l_coset_subset_G: "[| H ⊆ carrier G; x ∈ carrier G |] ==> x <# H ⊆ carrier G" by (auto simp add: l_coset_def subsetD) lemma (in group) l_coset_swap: "[|y ∈ x <# H; x ∈ carrier G; subgroup H G|] ==> x ∈ y <# H" proof (simp add: l_coset_def) assume "∃h∈H. y = x ⊗ h" and x: "x ∈ carrier G" and sb: "subgroup H G" then obtain h' where h': "h' ∈ H & x ⊗ h' = y" by blast show "∃h∈H. x = y ⊗ h" proof show "x = y ⊗ inv h'" using h' x sb by (auto simp add: m_assoc subgroup.subset [THEN subsetD]) show "inv h' ∈ H" using h' sb by (auto simp add: subgroup.subset [THEN subsetD] subgroup.m_inv_closed) qed qed lemma (in group) l_coset_carrier: "[| y ∈ x <# H; x ∈ carrier G; subgroup H G |] ==> y ∈ carrier G" by (auto simp add: l_coset_def m_assoc subgroup.subset [THEN subsetD] subgroup.m_closed) lemma (in group) l_repr_imp_subset: assumes y: "y ∈ x <# H" and x: "x ∈ carrier G" and sb: "subgroup H G" shows "y <# H ⊆ x <# H" proof - from y obtain h' where "h' ∈ H" "x ⊗ h' = y" by (auto simp add: l_coset_def) thus ?thesis using x sb by (auto simp add: l_coset_def m_assoc subgroup.subset [THEN subsetD] subgroup.m_closed) qed lemma (in group) l_repr_independence: assumes y: "y ∈ x <# H" and x: "x ∈ carrier G" and sb: "subgroup H G" shows "x <# H = y <# H" proof show "x <# H ⊆ y <# H" by (rule l_repr_imp_subset, (blast intro: l_coset_swap l_coset_carrier y x sb)+) show "y <# H ⊆ x <# H" by (rule l_repr_imp_subset [OF y x sb]) qed lemma (in group) setmult_subset_G: "[|H ⊆ carrier G; K ⊆ carrier G|] ==> H <#> K ⊆ carrier G" by (auto simp add: set_mult_def subsetD) lemma (in group) subgroup_mult_id: "subgroup H G ==> H <#> H = H" apply (auto simp add: subgroup.m_closed set_mult_def Sigma_def image_def) apply (rule_tac x = x in bexI) apply (rule bexI [of _ "\<one>"]) apply (auto simp add: subgroup.m_closed subgroup.one_closed r_one subgroup.subset [THEN subsetD]) done subsubsection {* Set of inverses of an @{text r_coset}. *} lemma (in normal) rcos_inv: assumes x: "x ∈ carrier G" shows "set_inv (H #> x) = H #> (inv x)" proof (simp add: r_coset_def SET_INV_def x inv_mult_group, safe) fix h assume "h ∈ H" show "inv x ⊗ inv h ∈ (\<Union>j∈H. {j ⊗ inv x})" proof show "inv x ⊗ inv h ⊗ x ∈ H" by (simp add: inv_op_closed1 prems) show "inv x ⊗ inv h ∈ {inv x ⊗ inv h ⊗ x ⊗ inv x}" by (simp add: prems m_assoc) qed next fix h assume "h ∈ H" show "h ⊗ inv x ∈ (\<Union>j∈H. {inv x ⊗ inv j})" proof show "x ⊗ inv h ⊗ inv x ∈ H" by (simp add: inv_op_closed2 prems) show "h ⊗ inv x ∈ {inv x ⊗ inv (x ⊗ inv h ⊗ inv x)}" by (simp add: prems m_assoc [symmetric] inv_mult_group) qed qed subsubsection {*Theorems for @{text "<#>"} with @{text "#>"} or @{text "<#"}.*} lemma (in group) setmult_rcos_assoc: "[|H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G|] ==> H <#> (K #> x) = (H <#> K) #> x" by (force simp add: r_coset_def set_mult_def m_assoc) lemma (in group) rcos_assoc_lcos: "[|H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G|] ==> (H #> x) <#> K = H <#> (x <# K)" by (force simp add: r_coset_def l_coset_def set_mult_def m_assoc) lemma (in normal) rcos_mult_step1: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y" by (simp add: setmult_rcos_assoc subset r_coset_subset_G l_coset_subset_G rcos_assoc_lcos) lemma (in normal) rcos_mult_step2: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y" by (insert coset_eq, simp add: normal_def) lemma (in normal) rcos_mult_step3: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H <#> (H #> x)) #> y = H #> (x ⊗ y)" by (simp add: setmult_rcos_assoc coset_mult_assoc subgroup_mult_id subset prems) lemma (in normal) rcos_sum: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H #> x) <#> (H #> y) = H #> (x ⊗ y)" by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3) lemma (in normal) rcosets_mult_eq: "M ∈ rcosets H ==> H <#> M = M" -- {* generalizes @{text subgroup_mult_id} *} by (auto simp add: RCOSETS_def subset setmult_rcos_assoc subgroup_mult_id prems) subsubsection{*An Equivalence Relation*} constdefs (structure G) r_congruent :: "[('a,'b)monoid_scheme, 'a set] => ('a*'a)set" ("rcong\<index> _") "rcong H ≡ {(x,y). x ∈ carrier G & y ∈ carrier G & inv x ⊗ y ∈ H}" lemma (in subgroup) equiv_rcong: includes group G shows "equiv (carrier G) (rcong H)" proof (intro equiv.intro) show "refl (carrier G) (rcong H)" by (auto simp add: r_congruent_def refl_def) next show "sym (rcong H)" proof (simp add: r_congruent_def sym_def, clarify) fix x y assume [simp]: "x ∈ carrier G" "y ∈ carrier G" and "inv x ⊗ y ∈ H" hence "inv (inv x ⊗ y) ∈ H" by (simp add: m_inv_closed) thus "inv y ⊗ x ∈ H" by (simp add: inv_mult_group) qed next show "trans (rcong H)" proof (simp add: r_congruent_def trans_def, clarify) fix x y z assume [simp]: "x ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G" and "inv x ⊗ y ∈ H" and "inv y ⊗ z ∈ H" hence "(inv x ⊗ y) ⊗ (inv y ⊗ z) ∈ H" by simp hence "inv x ⊗ (y ⊗ inv y) ⊗ z ∈ H" by (simp add: m_assoc del: r_inv) thus "inv x ⊗ z ∈ H" by simp qed qed text{*Equivalence classes of @{text rcong} correspond to left cosets. Was there a mistake in the definitions? I'd have expected them to correspond to right cosets.*} (* CB: This is correct, but subtle. We call H #> a the right coset of a relative to H. According to Jacobson, this is what the majority of group theory literature does. He then defines the notion of congruence relation ~ over monoids as equivalence relation with a ~ a' & b ~ b' ==> a*b ~ a'*b'. Our notion of right congruence induced by K: rcong K appears only in the context where K is a normal subgroup. Jacobson doesn't name it. But in this context left and right cosets are identical. *) lemma (in subgroup) l_coset_eq_rcong: includes group G assumes a: "a ∈ carrier G" shows "a <# H = rcong H `` {a}" by (force simp add: r_congruent_def l_coset_def m_assoc [symmetric] a ) subsubsection{*Two distinct right cosets are disjoint*} lemma (in group) rcos_equation: includes subgroup H G shows "[|ha ⊗ a = h ⊗ b; a ∈ carrier G; b ∈ carrier G; h ∈ H; ha ∈ H; hb ∈ H|] ==> hb ⊗ a ∈ (\<Union>h∈H. {h ⊗ b})" apply (rule UN_I [of "hb ⊗ ((inv ha) ⊗ h)"]) apply (simp add: ); apply (simp add: m_assoc transpose_inv) done lemma (in group) rcos_disjoint: includes subgroup H G shows "[|a ∈ rcosets H; b ∈ rcosets H; a≠b|] ==> a ∩ b = {}" apply (simp add: RCOSETS_def r_coset_def) apply (blast intro: rcos_equation prems sym) done subsection {*Order of a Group and Lagrange's Theorem*} constdefs order :: "('a, 'b) monoid_scheme => nat" "order S ≡ card (carrier S)" lemma (in group) rcos_self: includes subgroup shows "x ∈ carrier G ==> x ∈ H #> x" apply (simp add: r_coset_def) apply (rule_tac x="\<one>" in bexI) apply (auto simp add: ); done lemma (in group) rcosets_part_G: includes subgroup shows "\<Union>(rcosets H) = carrier G" apply (rule equalityI) apply (force simp add: RCOSETS_def r_coset_def) apply (auto simp add: RCOSETS_def intro: rcos_self prems) done lemma (in group) cosets_finite: "[|c ∈ rcosets H; H ⊆ carrier G; finite (carrier G)|] ==> finite c" apply (auto simp add: RCOSETS_def) apply (simp add: r_coset_subset_G [THEN finite_subset]) done text{*The next two lemmas support the proof of @{text card_cosets_equal}.*} lemma (in group) inj_on_f: "[|H ⊆ carrier G; a ∈ carrier G|] ==> inj_on (λy. y ⊗ inv a) (H #> a)" apply (rule inj_onI) apply (subgoal_tac "x ∈ carrier G & y ∈ carrier G") prefer 2 apply (blast intro: r_coset_subset_G [THEN subsetD]) apply (simp add: subsetD) done lemma (in group) inj_on_g: "[|H ⊆ carrier G; a ∈ carrier G|] ==> inj_on (λy. y ⊗ a) H" by (force simp add: inj_on_def subsetD) lemma (in group) card_cosets_equal: "[|c ∈ rcosets H; H ⊆ carrier G; finite(carrier G)|] ==> card c = card H" apply (auto simp add: RCOSETS_def) apply (rule card_bij_eq) apply (rule inj_on_f, assumption+) apply (force simp add: m_assoc subsetD r_coset_def) apply (rule inj_on_g, assumption+) apply (force simp add: m_assoc subsetD r_coset_def) txt{*The sets @{term "H #> a"} and @{term "H"} are finite.*} apply (simp add: r_coset_subset_G [THEN finite_subset]) apply (blast intro: finite_subset) done lemma (in group) rcosets_subset_PowG: "subgroup H G ==> rcosets H ⊆ Pow(carrier G)" apply (simp add: RCOSETS_def) apply (blast dest: r_coset_subset_G subgroup.subset) done theorem (in group) lagrange: "[|finite(carrier G); subgroup H G|] ==> card(rcosets H) * card(H) = order(G)" apply (simp (no_asm_simp) add: order_def rcosets_part_G [symmetric]) apply (subst mult_commute) apply (rule card_partition) apply (simp add: rcosets_subset_PowG [THEN finite_subset]) apply (simp add: rcosets_part_G) apply (simp add: card_cosets_equal subgroup.subset) apply (simp add: rcos_disjoint) done subsection {*Quotient Groups: Factorization of a Group*} constdefs FactGroup :: "[('a,'b) monoid_scheme, 'a set] => ('a set) monoid" (infixl "Mod" 65) --{*Actually defined for groups rather than monoids*} "FactGroup G H ≡ (|carrier = rcosetsG H, mult = set_mult G, one = H|))," lemma (in normal) setmult_closed: "[|K1 ∈ rcosets H; K2 ∈ rcosets H|] ==> K1 <#> K2 ∈ rcosets H" by (auto simp add: rcos_sum RCOSETS_def) lemma (in normal) setinv_closed: "K ∈ rcosets H ==> set_inv K ∈ rcosets H" by (auto simp add: rcos_inv RCOSETS_def) lemma (in normal) rcosets_assoc: "[|M1 ∈ rcosets H; M2 ∈ rcosets H; M3 ∈ rcosets H|] ==> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)" by (auto simp add: RCOSETS_def rcos_sum m_assoc) lemma (in subgroup) subgroup_in_rcosets: includes group G shows "H ∈ rcosets H" proof - have "H #> \<one> = H" by (rule coset_join2, auto) then show ?thesis by (auto simp add: RCOSETS_def) qed lemma (in normal) rcosets_inv_mult_group_eq: "M ∈ rcosets H ==> set_inv M <#> M = H" by (auto simp add: RCOSETS_def rcos_inv rcos_sum subgroup.subset prems) theorem (in normal) factorgroup_is_group: "group (G Mod H)" apply (simp add: FactGroup_def) apply (rule groupI) apply (simp add: setmult_closed) apply (simp add: normal_imp_subgroup subgroup_in_rcosets [OF is_group]) apply (simp add: restrictI setmult_closed rcosets_assoc) apply (simp add: normal_imp_subgroup subgroup_in_rcosets rcosets_mult_eq) apply (auto dest: rcosets_inv_mult_group_eq simp add: setinv_closed) done lemma mult_FactGroup [simp]: "X ⊗(G Mod H) X' = X <#>G X'" by (simp add: FactGroup_def) lemma (in normal) inv_FactGroup: "X ∈ carrier (G Mod H) ==> invG Mod H X = set_inv X" apply (rule group.inv_equality [OF factorgroup_is_group]) apply (simp_all add: FactGroup_def setinv_closed rcosets_inv_mult_group_eq) done text{*The coset map is a homomorphism from @{term G} to the quotient group @{term "G Mod H"}*} lemma (in normal) r_coset_hom_Mod: "(λa. H #> a) ∈ hom G (G Mod H)" by (auto simp add: FactGroup_def RCOSETS_def Pi_def hom_def rcos_sum) subsection{*The First Isomorphism Theorem*} text{*The quotient by the kernel of a homomorphism is isomorphic to the range of that homomorphism.*} constdefs kernel :: "('a, 'm) monoid_scheme => ('b, 'n) monoid_scheme => ('a => 'b) => 'a set" --{*the kernel of a homomorphism*} "kernel G H h ≡ {x. x ∈ carrier G & h x = \<one>H}"; lemma (in group_hom) subgroup_kernel: "subgroup (kernel G H h) G" apply (rule subgroup.intro) apply (auto simp add: kernel_def group.intro prems) done text{*The kernel of a homomorphism is a normal subgroup*} lemma (in group_hom) normal_kernel: "(kernel G H h) \<lhd> G" apply (simp add: group.normal_inv_iff subgroup_kernel group.intro prems) apply (simp add: kernel_def) done lemma (in group_hom) FactGroup_nonempty: assumes X: "X ∈ carrier (G Mod kernel G H h)" shows "X ≠ {}" proof - from X obtain g where "g ∈ carrier G" and "X = kernel G H h #> g" by (auto simp add: FactGroup_def RCOSETS_def) thus ?thesis by (auto simp add: kernel_def r_coset_def image_def intro: hom_one) qed lemma (in group_hom) FactGroup_contents_mem: assumes X: "X ∈ carrier (G Mod (kernel G H h))" shows "contents (h`X) ∈ carrier H" proof - from X obtain g where g: "g ∈ carrier G" and "X = kernel G H h #> g" by (auto simp add: FactGroup_def RCOSETS_def) hence "h ` X = {h g}" by (auto simp add: kernel_def r_coset_def image_def g) thus ?thesis by (auto simp add: g) qed lemma (in group_hom) FactGroup_hom: "(λX. contents (h`X)) ∈ hom (G Mod (kernel G H h)) H" apply (simp add: hom_def FactGroup_contents_mem normal.factorgroup_is_group [OF normal_kernel] group.axioms monoid.m_closed) proof (simp add: hom_def funcsetI FactGroup_contents_mem, intro ballI) fix X and X' assume X: "X ∈ carrier (G Mod kernel G H h)" and X': "X' ∈ carrier (G Mod kernel G H h)" then obtain g and g' where "g ∈ carrier G" and "g' ∈ carrier G" and "X = kernel G H h #> g" and "X' = kernel G H h #> g'" by (auto simp add: FactGroup_def RCOSETS_def) hence all: "∀x∈X. h x = h g" "∀x∈X'. h x = h g'" and Xsub: "X ⊆ carrier G" and X'sub: "X' ⊆ carrier G" by (force simp add: kernel_def r_coset_def image_def)+ hence "h ` (X <#> X') = {h g ⊗H h g'}" using X X' by (auto dest!: FactGroup_nonempty simp add: set_mult_def image_eq_UN subsetD [OF Xsub] subsetD [OF X'sub]) thus "contents (h ` (X <#> X')) = contents (h ` X) ⊗H contents (h ` X')" by (simp add: all image_eq_UN FactGroup_nonempty X X') qed text{*Lemma for the following injectivity result*} lemma (in group_hom) FactGroup_subset: "[|g ∈ carrier G; g' ∈ carrier G; h g = h g'|] ==> kernel G H h #> g ⊆ kernel G H h #> g'" apply (clarsimp simp add: kernel_def r_coset_def image_def); apply (rename_tac y) apply (rule_tac x="y ⊗ g ⊗ inv g'" in exI) apply (simp add: G.m_assoc); done lemma (in group_hom) FactGroup_inj_on: "inj_on (λX. contents (h ` X)) (carrier (G Mod kernel G H h))" proof (simp add: inj_on_def, clarify) fix X and X' assume X: "X ∈ carrier (G Mod kernel G H h)" and X': "X' ∈ carrier (G Mod kernel G H h)" then obtain g and g' where gX: "g ∈ carrier G" "g' ∈ carrier G" "X = kernel G H h #> g" "X' = kernel G H h #> g'" by (auto simp add: FactGroup_def RCOSETS_def) hence all: "∀x∈X. h x = h g" "∀x∈X'. h x = h g'" by (force simp add: kernel_def r_coset_def image_def)+ assume "contents (h ` X) = contents (h ` X')" hence h: "h g = h g'" by (simp add: image_eq_UN all FactGroup_nonempty X X') show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX) qed text{*If the homomorphism @{term h} is onto @{term H}, then so is the homomorphism from the quotient group*} lemma (in group_hom) FactGroup_onto: assumes h: "h ` carrier G = carrier H" shows "(λX. contents (h ` X)) ` carrier (G Mod kernel G H h) = carrier H" proof show "(λX. contents (h ` X)) ` carrier (G Mod kernel G H h) ⊆ carrier H" by (auto simp add: FactGroup_contents_mem) show "carrier H ⊆ (λX. contents (h ` X)) ` carrier (G Mod kernel G H h)" proof fix y assume y: "y ∈ carrier H" with h obtain g where g: "g ∈ carrier G" "h g = y" by (blast elim: equalityE); hence "(\<Union>x∈kernel G H h #> g. {h x}) = {y}" by (auto simp add: y kernel_def r_coset_def) with g show "y ∈ (λX. contents (h ` X)) ` carrier (G Mod kernel G H h)" by (auto intro!: bexI simp add: FactGroup_def RCOSETS_def image_eq_UN) qed qed text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*} theorem (in group_hom) FactGroup_iso: "h ` carrier G = carrier H ==> (λX. contents (h`X)) ∈ (G Mod (kernel G H h)) ≅ H" by (simp add: iso_def FactGroup_hom FactGroup_inj_on bij_betw_def FactGroup_onto) end
lemma coset_mult_assoc:
[| group G; M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> M #>G g #>G h = M #>G g ⊗G h
lemma coset_mult_one:
[| group G; M ⊆ carrier G |] ==> M #>G \<one>G = M
lemma coset_mult_inv1:
[| group G; M #>G x ⊗G invG y = M; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |] ==> M #>G x = M #>G y
lemma coset_mult_inv2:
[| group G; M #>G x = M #>G y; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |] ==> M #>G x ⊗G invG y = M
lemma coset_join1:
[| group G; H #>G x = H; x ∈ carrier G; subgroup H G |] ==> x ∈ H
lemma solve_equation:
[| group G; subgroup H G; x ∈ H; y ∈ H |] ==> ∃h∈H. y = h ⊗G x
lemma repr_independence:
[| group G; y ∈ H #>G x; x ∈ carrier G; subgroup H G |] ==> H #>G x = H #>G y
lemma coset_join2:
[| group G; x ∈ carrier G; subgroup H G; x ∈ H |] ==> H #>G x = H
lemma r_coset_subset_G:
[| group G; H ⊆ carrier G; x ∈ carrier G |] ==> H #>G x ⊆ carrier G
lemma rcosI:
[| group G; h ∈ H; H ⊆ carrier G; x ∈ carrier G |] ==> h ⊗G x ∈ H #>G x
lemma rcosetsI:
[| group G; H ⊆ carrier G; x ∈ carrier G |] ==> H #>G x ∈ rcosetsG H
lemma transpose_inv:
[| group G; x ⊗G y = z; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> invG x ⊗G z = y
lemma rcos_self:
[| group G; x ∈ carrier G; subgroup H G |] ==> x ∈ H #>G x
lemma normal_imp_subgroup:
H \<lhd> G ==> subgroup H G
lemma normalI:
[| group G; subgroup H G; ∀x∈carrier G. H #>G x = x <#G H |] ==> H \<lhd> G
lemma inv_op_closed1:
[| H \<lhd> G; x ∈ carrier G; h ∈ H |] ==> invG x ⊗G h ⊗G x ∈ H
lemma inv_op_closed2:
[| H \<lhd> G; x ∈ carrier G; h ∈ H |] ==> x ⊗G h ⊗G invG x ∈ H
lemma normal_inv_iff:
group G ==> N \<lhd> G = (subgroup N G ∧ (∀x∈carrier G. ∀h∈N. x ⊗G h ⊗G invG x ∈ N))
lemma lcos_m_assoc:
[| group G; M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> g <#G (h <#G M) = g ⊗G h <#G M
lemma lcos_mult_one:
[| group G; M ⊆ carrier G |] ==> \<one>G <#G M = M
lemma l_coset_subset_G:
[| group G; H ⊆ carrier G; x ∈ carrier G |] ==> x <#G H ⊆ carrier G
lemma l_coset_swap:
[| group G; y ∈ x <#G H; x ∈ carrier G; subgroup H G |] ==> x ∈ y <#G H
lemma l_coset_carrier:
[| group G; y ∈ x <#G H; x ∈ carrier G; subgroup H G |] ==> y ∈ carrier G
lemma l_repr_imp_subset:
[| group G; y ∈ x <#G H; x ∈ carrier G; subgroup H G |] ==> y <#G H ⊆ x <#G H
lemma l_repr_independence:
[| group G; y ∈ x <#G H; x ∈ carrier G; subgroup H G |] ==> x <#G H = y <#G H
lemma setmult_subset_G:
[| group G; H ⊆ carrier G; K ⊆ carrier G |] ==> H <#>G K ⊆ carrier G
lemma subgroup_mult_id:
[| group G; subgroup H G |] ==> H <#>G H = H
lemma rcos_inv:
[| H \<lhd> G; x ∈ carrier G |] ==> set_invG (H #>G x) = H #>G invG x
lemma setmult_rcos_assoc:
[| group G; H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G |] ==> H <#>G (K #>G x) = H <#>G K #>G x
lemma rcos_assoc_lcos:
[| group G; H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G |] ==> H #>G x <#>G K = H <#>G (x <#G K)
lemma rcos_mult_step1:
[| H \<lhd> G; x ∈ carrier G; y ∈ carrier G |] ==> H #>G x <#>G (H #>G y) = H <#>G (x <#G H) #>G y
lemma rcos_mult_step2:
[| H \<lhd> G; x ∈ carrier G; y ∈ carrier G |] ==> H <#>G (x <#G H) #>G y = H <#>G (H #>G x) #>G y
lemma rcos_mult_step3:
[| H \<lhd> G; x ∈ carrier G; y ∈ carrier G |] ==> H <#>G (H #>G x) #>G y = H #>G x ⊗G y
lemma rcos_sum:
[| H \<lhd> G; x ∈ carrier G; y ∈ carrier G |] ==> H #>G x <#>G (H #>G y) = H #>G x ⊗G y
lemma rcosets_mult_eq:
[| H \<lhd> G; M ∈ rcosetsG H |] ==> H <#>G M = M
lemma equiv_rcong:
[| subgroup H G; group G |] ==> equiv (carrier G) rcongG H
lemma l_coset_eq_rcong:
[| subgroup H G; group G; a ∈ carrier G |] ==> a <#G H = rcongG H `` {a}
lemma rcos_equation:
[| group G; subgroup H G; ha ⊗G a = h ⊗G b; a ∈ carrier G; b ∈ carrier G; h ∈ H; ha ∈ H; hb ∈ H |] ==> hb ⊗G a ∈ (UN h:H. {h ⊗G b})
lemma rcos_disjoint:
[| group G; subgroup H G; a ∈ rcosetsG H; b ∈ rcosetsG H; a ≠ b |] ==> a ∩ b = {}
lemma rcos_self:
[| group G; subgroup H G; x ∈ carrier G |] ==> x ∈ H #>G x
lemma rcosets_part_G:
[| group G; subgroup H G |] ==> Union (rcosetsG H) = carrier G
lemma cosets_finite:
[| group G; c ∈ rcosetsG H; H ⊆ carrier G; finite (carrier G) |] ==> finite c
lemma inj_on_f:
[| group G; H ⊆ carrier G; a ∈ carrier G |] ==> inj_on (%y. y ⊗G invG a) (H #>G a)
lemma inj_on_g:
[| group G; H ⊆ carrier G; a ∈ carrier G |] ==> inj_on (%y. y ⊗G a) H
lemma card_cosets_equal:
[| group G; c ∈ rcosetsG H; H ⊆ carrier G; finite (carrier G) |] ==> card c = card H
lemma rcosets_subset_PowG:
[| group G; subgroup H G |] ==> rcosetsG H ⊆ Pow (carrier G)
theorem lagrange:
[| group G; finite (carrier G); subgroup H G |] ==> card (rcosetsG H) * card H = order G
lemma setmult_closed:
[| H \<lhd> G; K1.0 ∈ rcosetsG H; K2.0 ∈ rcosetsG H |] ==> K1.0 <#>G K2.0 ∈ rcosetsG H
lemma setinv_closed:
[| H \<lhd> G; K ∈ rcosetsG H |] ==> set_invG K ∈ rcosetsG H
lemma rcosets_assoc:
[| H \<lhd> G; M1.0 ∈ rcosetsG H; M2.0 ∈ rcosetsG H; M3.0 ∈ rcosetsG H |] ==> M1.0 <#>G M2.0 <#>G M3.0 = M1.0 <#>G (M2.0 <#>G M3.0)
lemma subgroup_in_rcosets:
[| subgroup H G; group G |] ==> H ∈ rcosetsG H
lemma rcosets_inv_mult_group_eq:
[| H \<lhd> G; M ∈ rcosetsG H |] ==> set_invG M <#>G M = H
theorem factorgroup_is_group:
H \<lhd> G ==> group (G Mod H)
lemma mult_FactGroup:
X ⊗G Mod H X' = X <#>G X'
lemma inv_FactGroup:
[| H \<lhd> G; X ∈ carrier (G Mod H) |] ==> invG Mod H X = set_invG X
lemma r_coset_hom_Mod:
H \<lhd> G ==> op #>G H ∈ hom G (G Mod H)
lemma subgroup_kernel:
group_hom G H h ==> subgroup (kernel G H h) G
lemma normal_kernel:
group_hom G H h ==> kernel G H h \<lhd> G
lemma FactGroup_nonempty:
[| group_hom G H h; X ∈ carrier (G Mod kernel G H h) |] ==> X ≠ {}
lemma FactGroup_contents_mem:
[| group_hom G H h; X ∈ carrier (G Mod kernel G H h) |] ==> contents (h ` X) ∈ carrier H
lemma FactGroup_hom:
group_hom G H h ==> (%X. contents (h ` X)) ∈ hom (G Mod kernel G H h) H
lemma FactGroup_subset:
[| group_hom G H h; g ∈ carrier G; g' ∈ carrier G; h g = h g' |] ==> kernel G H h #>G g ⊆ kernel G H h #>G g'
lemma FactGroup_inj_on:
group_hom G H h ==> inj_on (%X. contents (h ` X)) (carrier (G Mod kernel G H h))
lemma FactGroup_onto:
[| group_hom G H h; h ` carrier G = carrier H |] ==> (%X. contents (h ` X)) ` carrier (G Mod kernel G H h) = carrier H
theorem FactGroup_iso:
[| group_hom G H h; h ` carrier G = carrier H |] ==> (%X. contents (h ` X)) ∈ G Mod kernel G H h ≅ H