(* Title: ZF/ex/Group.thy Id: $Id: Group.thy,v 1.7 2007/05/10 22:43:45 wenzelm Exp $ *) header {* Groups *} theory Group imports Main begin text{*Based on work by Clemens Ballarin, Florian Kammueller, L C Paulson and Markus Wenzel.*} subsection {* Monoids *} (*First, we must simulate a record declaration: record monoid = carrier :: i mult :: "[i,i] => i" (infixl "·\<index>" 70) one :: i ("\<one>\<index>") *) definition carrier :: "i => i" where "carrier(M) == fst(M)" definition mmult :: "[i, i, i] => i" (infixl "·\<index>" 70) where "mmult(M,x,y) == fst(snd(M)) ` <x,y>" definition one :: "i => i" ("\<one>\<index>") where "one(M) == fst(snd(snd(M)))" definition update_carrier :: "[i,i] => i" where "update_carrier(M,A) == <A,snd(M)>" definition m_inv :: "i => i => i" ("inv\<index> _" [81] 80) where "invG x == (THE y. y ∈ carrier(G) & y ·G x = \<one>G & x ·G y = \<one>G)" locale monoid = struct G + assumes m_closed [intro, simp]: "[|x ∈ carrier(G); y ∈ carrier(G)|] ==> x · y ∈ carrier(G)" and m_assoc: "[|x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G)|] ==> (x · y) · z = x · (y · z)" and one_closed [intro, simp]: "\<one> ∈ carrier(G)" and l_one [simp]: "x ∈ carrier(G) ==> \<one> · x = x" and r_one [simp]: "x ∈ carrier(G) ==> x · \<one> = x" text{*Simulating the record*} lemma carrier_eq [simp]: "carrier(<A,Z>) = A" by (simp add: carrier_def) lemma mult_eq [simp]: "mmult(<A,M,Z>, x, y) = M ` <x,y>" by (simp add: mmult_def) lemma one_eq [simp]: "one(<A,M,I,Z>) = I" by (simp add: one_def) lemma update_carrier_eq [simp]: "update_carrier(<A,Z>,B) = <B,Z>" by (simp add: update_carrier_def) lemma carrier_update_carrier [simp]: "carrier(update_carrier(M,B)) = B" by (simp add: update_carrier_def) lemma mult_update_carrier [simp]: "mmult(update_carrier(M,B),x,y) = mmult(M,x,y)" by (simp add: update_carrier_def mmult_def) lemma one_update_carrier [simp]: "one(update_carrier(M,B)) = one(M)" by (simp add: update_carrier_def one_def) lemma (in monoid) inv_unique: assumes eq: "y · x = \<one>" "x · y' = \<one>" and G: "x ∈ carrier(G)" "y ∈ carrier(G)" "y' ∈ carrier(G)" shows "y = y'" proof - from G eq have "y = y · (x · y')" by simp also from G have "... = (y · x) · y'" by (simp add: m_assoc) also from G eq have "... = y'" by simp finally show ?thesis . qed text {* A group is a monoid all of whose elements are invertible. *} locale group = monoid + assumes inv_ex: "!!x. x ∈ carrier(G) ==> ∃y ∈ carrier(G). y · x = \<one> & x · y = \<one>" lemma (in group) is_group [simp]: "group(G)" by fact theorem groupI: includes struct G assumes m_closed [simp]: "!!x y. [|x ∈ carrier(G); y ∈ carrier(G)|] ==> x · y ∈ carrier(G)" and one_closed [simp]: "\<one> ∈ carrier(G)" and m_assoc: "!!x y z. [|x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G)|] ==> (x · y) · z = x · (y · z)" and l_one [simp]: "!!x. x ∈ carrier(G) ==> \<one> · x = x" and l_inv_ex: "!!x. x ∈ carrier(G) ==> ∃y ∈ carrier(G). y · x = \<one>" shows "group(G)" proof - have l_cancel [simp]: "!!x y z. [|x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G)|] ==> (x · y = x · z) <-> (y = z)" proof fix x y z assume G: "x ∈ carrier(G)" "y ∈ carrier(G)" "z ∈ carrier(G)" { assume eq: "x · y = x · z" with G l_inv_ex obtain x_inv where xG: "x_inv ∈ carrier(G)" and l_inv: "x_inv · x = \<one>" by fast from G eq xG have "(x_inv · x) · y = (x_inv · x) · z" by (simp add: m_assoc) with G show "y = z" by (simp add: l_inv) next assume eq: "y = z" with G show "x · y = x · z" by simp } qed have r_one: "!!x. x ∈ carrier(G) ==> x · \<one> = x" proof - fix x assume x: "x ∈ carrier(G)" with l_inv_ex obtain x_inv where xG: "x_inv ∈ carrier(G)" and l_inv: "x_inv · x = \<one>" by fast from x xG have "x_inv · (x · \<one>) = x_inv · x" by (simp add: m_assoc [symmetric] l_inv) with x xG show "x · \<one> = x" by simp qed have inv_ex: "!!x. x ∈ carrier(G) ==> ∃y ∈ carrier(G). y · x = \<one> & x · y = \<one>" proof - fix x assume x: "x ∈ carrier(G)" with l_inv_ex obtain y where y: "y ∈ carrier(G)" and l_inv: "y · x = \<one>" by fast from x y have "y · (x · y) = y · \<one>" by (simp add: m_assoc [symmetric] l_inv r_one) with x y have r_inv: "x · y = \<one>" by simp from x y show "∃y ∈ carrier(G). y · x = \<one> & x · y = \<one>" by (fast intro: l_inv r_inv) qed show ?thesis by (blast intro: group.intro monoid.intro group_axioms.intro prems r_one inv_ex) qed lemma (in group) inv [simp]: "x ∈ carrier(G) ==> inv x ∈ carrier(G) & inv x · x = \<one> & x · inv x = \<one>" apply (frule inv_ex) apply (unfold Bex_def m_inv_def) apply (erule exE) apply (rule theI) apply (rule ex1I, assumption) apply (blast intro: inv_unique) done lemma (in group) inv_closed [intro!]: "x ∈ carrier(G) ==> inv x ∈ carrier(G)" by simp lemma (in group) l_inv: "x ∈ carrier(G) ==> inv x · x = \<one>" by simp lemma (in group) r_inv: "x ∈ carrier(G) ==> x · inv x = \<one>" by simp subsection {* Cancellation Laws and Basic Properties *} lemma (in group) l_cancel [simp]: assumes [simp]: "x ∈ carrier(G)" "y ∈ carrier(G)" "z ∈ carrier(G)" shows "(x · y = x · z) <-> (y = z)" proof assume eq: "x · y = x · z" hence "(inv x · x) · y = (inv x · x) · z" by (simp only: m_assoc inv_closed prems) thus "y = z" by simp next assume eq: "y = z" then show "x · y = x · z" by simp qed lemma (in group) r_cancel [simp]: assumes [simp]: "x ∈ carrier(G)" "y ∈ carrier(G)" "z ∈ carrier(G)" shows "(y · x = z · x) <-> (y = z)" proof assume eq: "y · x = z · x" then have "y · (x · inv x) = z · (x · inv x)" by (simp only: m_assoc [symmetric] inv_closed prems) thus "y = z" by simp next assume eq: "y = z" thus "y · x = z · x" by simp qed lemma (in group) inv_comm: assumes inv: "x · y = \<one>" and G: "x ∈ carrier(G)" "y ∈ carrier(G)" shows "y · x = \<one>" proof - from G have "x · y · x = x · \<one>" by (auto simp add: inv) with G show ?thesis by (simp del: r_one add: m_assoc) qed lemma (in group) inv_equality: "[|y · x = \<one>; x ∈ carrier(G); y ∈ carrier(G)|] ==> inv x = y" apply (simp add: m_inv_def) apply (rule the_equality) apply (simp add: inv_comm [of y x]) apply (rule r_cancel [THEN iffD1], auto) done lemma (in group) inv_one [simp]: "inv \<one> = \<one>" by (auto intro: inv_equality) lemma (in group) inv_inv [simp]: "x ∈ carrier(G) ==> inv (inv x) = x" by (auto intro: inv_equality) text{*This proof is by cancellation*} lemma (in group) inv_mult_group: "[|x ∈ carrier(G); y ∈ carrier(G)|] ==> inv (x · y) = inv y · inv x" proof - assume G: "x ∈ carrier(G)" "y ∈ carrier(G)" then have "inv (x · y) · (x · y) = (inv y · inv x) · (x · y)" by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric] l_inv) with G show ?thesis by (simp_all del: inv add: inv_closed) qed subsection {* Substructures *} locale subgroup = var H + struct G + assumes subset: "H ⊆ carrier(G)" and m_closed [intro, simp]: "[|x ∈ H; y ∈ H|] ==> x · y ∈ H" and one_closed [simp]: "\<one> ∈ H" and m_inv_closed [intro,simp]: "x ∈ H ==> inv x ∈ H" lemma (in subgroup) mem_carrier [simp]: "x ∈ H ==> x ∈ carrier(G)" using subset by blast lemma subgroup_imp_subset: "subgroup(H,G) ==> H ⊆ carrier(G)" by (rule subgroup.subset) lemma (in subgroup) group_axiomsI [intro]: includes group G shows "group_axioms (update_carrier(G,H))" by (force intro: group_axioms.intro l_inv r_inv) lemma (in subgroup) is_group [intro]: includes group G shows "group (update_carrier(G,H))" by (rule groupI) (auto intro: m_assoc l_inv mem_carrier) text {* Since @{term H} is nonempty, it contains some element @{term x}. Since it is closed under inverse, it contains @{text "inv x"}. Since it is closed under product, it contains @{text "x · inv x = \<one>"}. *} text {* Since @{term H} is nonempty, it contains some element @{term x}. Since it is closed under inverse, it contains @{text "inv x"}. Since it is closed under product, it contains @{text "x · inv x = \<one>"}. *} lemma (in group) one_in_subset: "[|H ⊆ carrier(G); H ≠ 0; ∀a ∈ H. inv a ∈ H; ∀a∈H. ∀b∈H. a · b ∈ H|] ==> \<one> ∈ H" by (force simp add: l_inv) text {* A characterization of subgroups: closed, non-empty subset. *} declare monoid.one_closed [simp] group.inv_closed [simp] monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp] lemma subgroup_nonempty: "~ subgroup(0,G)" by (blast dest: subgroup.one_closed) subsection {* Direct Products *} definition DirProdGroup :: "[i,i] => i" (infixr "\<Otimes>" 80) where "G \<Otimes> H == <carrier(G) × carrier(H), (λ<<g,h>, <g', h'>> ∈ (carrier(G) × carrier(H)) × (carrier(G) × carrier(H)). <g ·G g', h ·H h'>), <\<one>G, \<one>H>, 0>" lemma DirProdGroup_group: includes group G + group H shows "group (G \<Otimes> H)" by (force intro!: groupI G.m_assoc H.m_assoc G.l_inv H.l_inv simp add: DirProdGroup_def) lemma carrier_DirProdGroup [simp]: "carrier (G \<Otimes> H) = carrier(G) × carrier(H)" by (simp add: DirProdGroup_def) lemma one_DirProdGroup [simp]: "\<one>G \<Otimes> H = <\<one>G, \<one>H>" by (simp add: DirProdGroup_def) lemma mult_DirProdGroup [simp]: "[|g ∈ carrier(G); h ∈ carrier(H); g' ∈ carrier(G); h' ∈ carrier(H)|] ==> <g, h> ·G \<Otimes> H <g', h'> = <g ·G g', h ·H h'>" by (simp add: DirProdGroup_def) lemma inv_DirProdGroup [simp]: includes group G + group H assumes g: "g ∈ carrier(G)" and h: "h ∈ carrier(H)" shows "inv G \<Otimes> H <g, h> = <invG g, invH h>" apply (rule group.inv_equality [OF DirProdGroup_group]) apply (simp_all add: prems group.l_inv) done subsection {* Isomorphisms *} definition hom :: "[i,i] => i" where "hom(G,H) == {h ∈ carrier(G) -> carrier(H). (∀x ∈ carrier(G). ∀y ∈ carrier(G). h ` (x ·G y) = (h ` x) ·H (h ` y))}" lemma hom_mult: "[|h ∈ hom(G,H); x ∈ carrier(G); y ∈ carrier(G)|] ==> h ` (x ·G y) = h ` x ·H h ` y" by (simp add: hom_def) lemma hom_closed: "[|h ∈ hom(G,H); x ∈ carrier(G)|] ==> h ` x ∈ carrier(H)" by (auto simp add: hom_def) lemma (in group) hom_compose: "[|h ∈ hom(G,H); i ∈ hom(H,I)|] ==> i O h ∈ hom(G,I)" by (force simp add: hom_def comp_fun) lemma hom_is_fun: "h ∈ hom(G,H) ==> h ∈ carrier(G) -> carrier(H)" by (simp add: hom_def) subsection {* Isomorphisms *} definition iso :: "[i,i] => i" (infixr "≅" 60) where "G ≅ H == hom(G,H) ∩ bij(carrier(G), carrier(H))" lemma (in group) iso_refl: "id(carrier(G)) ∈ G ≅ G" by (simp add: iso_def hom_def id_type id_bij) lemma (in group) iso_sym: "h ∈ G ≅ H ==> converse(h) ∈ H ≅ G" apply (simp add: iso_def bij_converse_bij, clarify) apply (subgoal_tac "converse(h) ∈ carrier(H) -> carrier(G)") prefer 2 apply (simp add: bij_converse_bij bij_is_fun) apply (auto intro: left_inverse_eq [of _ "carrier(G)" "carrier(H)"] simp add: hom_def bij_is_inj right_inverse_bij); done lemma (in group) iso_trans: "[|h ∈ G ≅ H; i ∈ H ≅ I|] ==> i O h ∈ G ≅ I" by (auto simp add: iso_def hom_compose comp_bij) lemma DirProdGroup_commute_iso: includes group G + group H shows "(λ<x,y> ∈ carrier(G \<Otimes> H). <y,x>) ∈ (G \<Otimes> H) ≅ (H \<Otimes> G)" by (auto simp add: iso_def hom_def inj_def surj_def bij_def) lemma DirProdGroup_assoc_iso: includes group G + group H + group I shows "(λ<<x,y>,z> ∈ carrier((G \<Otimes> H) \<Otimes> I). <x,<y,z>>) ∈ ((G \<Otimes> H) \<Otimes> I) ≅ (G \<Otimes> (H \<Otimes> I))" by (auto intro: lam_type simp add: iso_def hom_def inj_def surj_def bij_def) text{*Basis for homomorphism proofs: we assume two groups @{term G} and @term{H}, with a homomorphism @{term h} between them*} locale group_hom = group G + group H + var h + assumes homh: "h ∈ hom(G,H)" notes hom_mult [simp] = hom_mult [OF homh] and hom_closed [simp] = hom_closed [OF homh] and hom_is_fun [simp] = hom_is_fun [OF homh] lemma (in group_hom) one_closed [simp]: "h ` \<one> ∈ carrier(H)" by simp lemma (in group_hom) hom_one [simp]: "h ` \<one> = \<one>H" proof - have "h ` \<one> ·H \<one>H = (h ` \<one>) ·H (h ` \<one>)" by (simp add: hom_mult [symmetric] del: hom_mult) then show ?thesis by (simp del: r_one) qed lemma (in group_hom) inv_closed [simp]: "x ∈ carrier(G) ==> h ` (inv x) ∈ carrier(H)" by simp lemma (in group_hom) hom_inv [simp]: "x ∈ carrier(G) ==> h ` (inv x) = invH (h ` x)" proof - assume x: "x ∈ carrier(G)" then have "h ` x ·H h ` (inv x) = \<one>H" by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult) also from x have "... = h ` x ·H invH (h ` x)" by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult) finally have "h ` x ·H h ` (inv x) = h ` x ·H invH (h ` x)" . with x show ?thesis by (simp del: inv add: is_group) qed subsection {* Commutative Structures *} text {* Naming convention: multiplicative structures that are commutative are called \emph{commutative}, additive structures are called \emph{Abelian}. *} subsection {* Definition *} locale comm_monoid = monoid + assumes m_comm: "[|x ∈ carrier(G); y ∈ carrier(G)|] ==> x · y = y · x" lemma (in comm_monoid) m_lcomm: "[|x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G)|] ==> x · (y · z) = y · (x · z)" proof - assume xyz: "x ∈ carrier(G)" "y ∈ carrier(G)" "z ∈ carrier(G)" from xyz have "x · (y · z) = (x · y) · z" by (simp add: m_assoc) also from xyz have "... = (y · x) · z" by (simp add: m_comm) also from xyz have "... = y · (x · z)" by (simp add: m_assoc) finally show ?thesis . qed lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm locale comm_group = comm_monoid + group lemma (in comm_group) inv_mult: "[|x ∈ carrier(G); y ∈ carrier(G)|] ==> inv (x · y) = inv x · inv y" by (simp add: m_ac inv_mult_group) lemma (in group) subgroup_self: "subgroup (carrier(G),G)" by (simp add: subgroup_def prems) lemma (in group) subgroup_imp_group: "subgroup(H,G) ==> group (update_carrier(G,H))" by (simp add: subgroup.is_group) lemma (in group) subgroupI: assumes subset: "H ⊆ carrier(G)" and non_empty: "H ≠ 0" and inv: "!!a. a ∈ H ==> inv a ∈ H" and mult: "!!a b. [|a ∈ H; b ∈ H|] ==> a · b ∈ H" shows "subgroup(H,G)" proof (simp add: subgroup_def prems) show "\<one> ∈ H" by (rule one_in_subset) (auto simp only: prems) qed subsection {* Bijections of a Set, Permutation Groups, Automorphism Groups *} definition BijGroup :: "i=>i" where "BijGroup(S) == <bij(S,S), λ<g,f> ∈ bij(S,S) × bij(S,S). g O f, id(S), 0>" subsection {*Bijections Form a Group *} theorem group_BijGroup: "group(BijGroup(S))" apply (simp add: BijGroup_def) apply (rule groupI) apply (simp_all add: id_bij comp_bij comp_assoc) apply (simp add: id_bij bij_is_fun left_comp_id [of _ S S] fun_is_rel) apply (blast intro: left_comp_inverse bij_is_inj bij_converse_bij) done subsection{*Automorphisms Form a Group*} lemma Bij_Inv_mem: "[|f ∈ bij(S,S); x ∈ S|] ==> converse(f) ` x ∈ S" by (blast intro: apply_funtype bij_is_fun bij_converse_bij) lemma inv_BijGroup: "f ∈ bij(S,S) ==> m_inv (BijGroup(S), f) = converse(f)" apply (rule group.inv_equality) apply (rule group_BijGroup) apply (simp_all add: BijGroup_def bij_converse_bij left_comp_inverse [OF bij_is_inj]) done lemma iso_is_bij: "h ∈ G ≅ H ==> h ∈ bij(carrier(G), carrier(H))" by (simp add: iso_def) definition auto :: "i=>i" where "auto(G) == iso(G,G)" definition AutoGroup :: "i=>i" where "AutoGroup(G) == update_carrier(BijGroup(carrier(G)), auto(G))" lemma (in group) id_in_auto: "id(carrier(G)) ∈ auto(G)" by (simp add: iso_refl auto_def) lemma (in group) subgroup_auto: "subgroup (auto(G)) (BijGroup (carrier(G)))" proof (rule subgroup.intro) show "auto(G) ⊆ carrier (BijGroup (carrier(G)))" by (auto simp add: auto_def BijGroup_def iso_def) next fix x y assume "x ∈ auto(G)" "y ∈ auto(G)" thus "x ·BijGroup (carrier(G)) y ∈ auto(G)" by (auto simp add: BijGroup_def auto_def iso_def bij_is_fun group.hom_compose comp_bij) next show "\<one>BijGroup (carrier(G)) ∈ auto(G)" by (simp add: BijGroup_def id_in_auto) next fix x assume "x ∈ auto(G)" thus "invBijGroup (carrier(G)) x ∈ auto(G)" by (simp add: auto_def inv_BijGroup iso_is_bij iso_sym) qed theorem (in group) AutoGroup: "group (AutoGroup(G))" by (simp add: AutoGroup_def subgroup.is_group subgroup_auto group_BijGroup) subsection{*Cosets and Quotient Groups*} definition r_coset :: "[i,i,i] => i" (infixl "#>\<index>" 60) where "H #>G a == \<Union>h∈H. {h ·G a}" definition l_coset :: "[i,i,i] => i" (infixl "<#\<index>" 60) where "a <#G H == \<Union>h∈H. {a ·G h}" definition RCOSETS :: "[i,i] => i" ("rcosets\<index> _" [81] 80) where "rcosetsG H == \<Union>a∈carrier(G). {H #>G a}" definition set_mult :: "[i,i,i] => i" (infixl "<#>\<index>" 60) where "H <#>G K == \<Union>h∈H. \<Union>k∈K. {h ·G k}" definition SET_INV :: "[i,i] => i" ("set'_inv\<index> _" [81] 80) where "set_invG H == \<Union>h∈H. {invG h}" locale normal = subgroup + group + assumes coset_eq: "(∀x ∈ carrier(G). H #> x = x <# H)" notation normal (infixl "\<lhd>" 60) 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) 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]) 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) 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 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 (rule UN_I) 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 (rule UN_I) 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) 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 (rule equalityI) 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 intro!: equalityI) fix h assume "h ∈ H" show "inv x · inv h ∈ (\<Union>j∈H. {j · inv x})" proof (rule UN_I) 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 (rule UN_I) 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 normal.axioms) 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 normal.axioms) subsubsection{*Two distinct right cosets are disjoint*} definition r_congruent :: "[i,i] => i" ("rcong\<index> _" [60] 60) where "rcongG H == {<x,y> ∈ carrier(G) * carrier(G). invG x ·G y ∈ H}" lemma (in subgroup) equiv_rcong: includes group G shows "equiv (carrier(G), rcong H)" proof (simp add: equiv_def, intro conjI) show "rcong H ⊆ carrier(G) × carrier(G)" by (auto simp add: r_congruent_def) next 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: 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.*} 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 Collect_image_eq) 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)"], simp) 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 = 0" 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*} definition order :: "i => i" where "order(S) == |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, auto) 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 subset_Finite]) done text{*More general than the HOL version, which also requires @{term G} to be finite.*} lemma (in group) card_cosets_equal: assumes H: "H ⊆ carrier(G)" shows "c ∈ rcosets H ==> |c| = |H|" proof (simp add: RCOSETS_def, clarify) fix a assume a: "a ∈ carrier(G)" show "|H #> a| = |H|" proof (rule eqpollI [THEN cardinal_cong]) show "H #> a \<lesssim> H" proof (simp add: lepoll_def, intro exI) show "(λy ∈ H#>a. y · inv a) ∈ inj(H #> a, H)" by (auto intro: lam_type simp add: inj_def r_coset_def m_assoc subsetD [OF H] a) qed show "H \<lesssim> H #> a" proof (simp add: lepoll_def, intro exI) show "(λy∈ H. y · a) ∈ inj(H, H #> a)" by (auto intro: lam_type simp add: inj_def r_coset_def subsetD [OF H] a) qed qed qed 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)|] ==> |rcosets H| #* |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 subset_Finite]) apply (simp add: rcosets_part_G) apply (simp add: card_cosets_equal [OF subgroup.subset]) apply (simp add: rcos_disjoint) done subsection {*Quotient Groups: Factorization of a Group*} definition FactGroup :: "[i,i] => i" (infixl "Mod" 65) where --{*Actually defined for groups rather than monoids*} "G Mod H == <rcosetsG H, λ<K1,K2> ∈ (rcosetsG H) × (rcosetsG H). K1 <#>G K2, H, 0>" 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" using _ `subgroup(H, G)` by (rule coset_join2) simp_all then show ?thesis by (auto simp add: RCOSETS_def intro: sym) 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 normal.axioms) 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) apply (simp add: 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 (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 ∈ carrier(G). H #> a) ∈ hom(G, G Mod H)" by (auto simp add: FactGroup_def RCOSETS_def hom_def rcos_sum intro: lam_type) subsection{*The First Isomorphism Theorem*} text{*The quotient by the kernel of a homomorphism is isomorphic to the range of that homomorphism.*} definition kernel :: "[i,i,i] => i" where --{*the kernel of a homomorphism*} "kernel(G,H,h) == {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 ≠ 0" 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_UN image_eq_UN [OF hom_is_fun] g) thus ?thesis by (auto simp add: g) qed lemma mult_FactGroup: "[|X ∈ carrier(G Mod H); X' ∈ carrier(G Mod H)|] ==> X ·(G Mod H) X' = X <#>G X'" by (simp add: FactGroup_def) lemma (in normal) FactGroup_m_closed: "[|X ∈ carrier(G Mod H); X' ∈ carrier(G Mod H)|] ==> X <#>G X' ∈ carrier(G Mod H)" by (simp add: FactGroup_def setmult_closed) lemma (in group_hom) FactGroup_hom: "(λX ∈ carrier(G Mod (kernel(G,H,h))). contents (h``X)) ∈ hom (G Mod (kernel(G,H,h)), H)" proof (simp add: hom_def FactGroup_contents_mem lam_type mult_FactGroup normal.FactGroup_m_closed [OF normal_kernel], 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 [OF hom_is_fun] image_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 [OF hom_is_fun] FactGroup_nonempty X X' Xsub X'sub) 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 bexI) apply (simp_all add: G.m_assoc) done lemma (in group_hom) FactGroup_inj: "(λX∈carrier (G Mod kernel(G,H,h)). contents (h `` X)) ∈ inj(carrier (G Mod kernel(G,H,h)), carrier(H))" proof (simp add: inj_def FactGroup_contents_mem lam_type, 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'" and Xsub: "X ⊆ carrier(G)" and X'sub: "X' ⊆ carrier(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: all image_eq_UN [OF hom_is_fun] FactGroup_nonempty X X' Xsub X'sub) show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX) qed lemma (in group_hom) kernel_rcoset_subset: assumes g: "g ∈ carrier(G)" shows "kernel(G,H,h) #> g ⊆ carrier (G)" by (auto simp add: g kernel_def r_coset_def) text{*If the homomorphism @{term h} is onto @{term H}, then so is the homomorphism from the quotient group*} lemma (in group_hom) FactGroup_surj: assumes h: "h ∈ surj(carrier(G), carrier(H))" shows "(λX∈carrier (G Mod kernel(G,H,h)). contents (h `` X)) ∈ surj(carrier (G Mod kernel(G,H,h)), carrier(H))" proof (simp add: surj_def FactGroup_contents_mem lam_type, clarify) fix y assume y: "y ∈ carrier(H)" with h obtain g where g: "g ∈ carrier(G)" "h ` g = y" by (auto simp add: surj_def) hence "(\<Union>x∈kernel(G,H,h) #> g. {h ` x}) = {y}" by (auto simp add: y kernel_def r_coset_def) with g show "∃x∈carrier(G Mod kernel(G, H, h)). contents(h `` x) = y" --{*The witness is @{term "kernel(G,H,h) #> g"}*} by (force simp add: FactGroup_def RCOSETS_def image_eq_UN [OF hom_is_fun] kernel_rcoset_subset) 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 ∈ surj(carrier(G), carrier(H)) ==> (λX∈carrier (G Mod kernel(G,H,h)). contents (h``X)) ∈ (G Mod (kernel(G,H,h))) ≅ H" by (simp add: iso_def FactGroup_hom FactGroup_inj bij_def FactGroup_surj) end
lemma carrier_eq:
carrier(〈A, Z〉) = A
lemma mult_eq:
x ·〈A, M, Z〉 y = M ` 〈x, y〉
lemma one_eq:
\<one>〈A, M, I, Z〉 = I
lemma update_carrier_eq:
update_carrier(〈A, Z〉, B) = 〈B, Z〉
lemma carrier_update_carrier:
carrier(update_carrier(M, B)) = B
lemma mult_update_carrier:
x ·update_carrier(M, B) y = x ·M y
lemma one_update_carrier:
\<one>update_carrier(M, B) = \<one>M
lemma inv_unique:
[| y · x = \<one>; x · y' = \<one>; x ∈ carrier(G); y ∈ carrier(G);
y' ∈ carrier(G) |]
==> y = y'
lemma is_group:
group(G)
theorem groupI:
[| !!x y. [| x ∈ carrier(G); y ∈ carrier(G) |] ==> x ·G y ∈ carrier(G);
\<one>G ∈ carrier(G);
!!x y z.
[| x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G) |]
==> x ·G y ·G z = x ·G (y ·G z);
!!x. x ∈ carrier(G) ==> \<one>G ·G x = x;
!!x. x ∈ carrier(G) ==> ∃y∈carrier(G). y ·G x = \<one>G |]
==> group(G)
lemma inv:
x ∈ carrier(G) ==> inv x ∈ carrier(G) ∧ inv x · x = \<one> ∧ x · inv x = \<one>
lemma inv_closed:
x ∈ carrier(G) ==> inv x ∈ carrier(G)
lemma l_inv:
x ∈ carrier(G) ==> inv x · x = \<one>
lemma r_inv:
x ∈ carrier(G) ==> x · inv x = \<one>
lemma l_cancel:
[| x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G) |] ==> x · y = x · z <-> y = z
lemma r_cancel:
[| x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G) |] ==> y · x = z · x <-> y = z
lemma inv_comm:
[| x · y = \<one>; x ∈ carrier(G); y ∈ carrier(G) |] ==> y · x = \<one>
lemma inv_equality:
[| y · x = \<one>; x ∈ carrier(G); y ∈ carrier(G) |] ==> inv x = y
lemma inv_one:
inv \<one> = \<one>
lemma inv_inv:
x ∈ carrier(G) ==> inv (inv x) = x
lemma inv_mult_group:
[| x ∈ carrier(G); y ∈ carrier(G) |] ==> inv (x · y) = inv y · inv x
lemma mem_carrier:
x ∈ H ==> x ∈ carrier(G)
lemma subgroup_imp_subset:
subgroup(H, G) ==> H ⊆ carrier(G)
lemma group_axiomsI:
group(G) ==> group_axioms(update_carrier(G, H))
lemma is_group:
group(G) ==> group(update_carrier(G, H))
lemma one_in_subset:
[| H ⊆ carrier(G); H ≠ 0; ∀a∈H. inv a ∈ H; ∀a∈H. ∀b∈H. a · b ∈ H |]
==> \<one> ∈ H
lemma subgroup_nonempty:
¬ subgroup(0, G)
lemma DirProdGroup_group:
[| group(G); group(H) |] ==> group(G \<Otimes> H)
lemma carrier_DirProdGroup:
carrier(G \<Otimes> H) = carrier(G) × carrier(H)
lemma one_DirProdGroup:
\<one>G \<Otimes> H = 〈\<one>G, \<one>H〉
lemma mult_DirProdGroup:
[| g ∈ carrier(G); h ∈ carrier(H); g' ∈ carrier(G); h' ∈ carrier(H) |]
==> 〈g, h〉 ·G \<Otimes> H 〈g', h'〉 = 〈g ·G g', h ·H h'〉
lemma inv_DirProdGroup:
[| group(G); group(H); g ∈ carrier(G); h ∈ carrier(H) |]
==> invG \<Otimes> H 〈g, h〉 = 〈invG g, invH h〉
lemma hom_mult:
[| h ∈ hom(G, H); x ∈ carrier(G); y ∈ carrier(G) |]
==> h ` (x ·G y) = h ` x ·H h ` y
lemma hom_closed:
[| h ∈ hom(G, H); x ∈ carrier(G) |] ==> h ` x ∈ carrier(H)
lemma hom_compose:
[| h ∈ hom(G, H); i ∈ hom(H, I) |] ==> i O h ∈ hom(G, I)
lemma hom_is_fun:
h ∈ hom(G, H) ==> h ∈ carrier(G) -> carrier(H)
lemma iso_refl:
id(carrier(G)) ∈ G ≅ G
lemma iso_sym:
h ∈ G ≅ H ==> converse(h) ∈ H ≅ G
lemma iso_trans:
[| h ∈ G ≅ H; i ∈ H ≅ I |] ==> i O h ∈ G ≅ I
lemma DirProdGroup_commute_iso:
[| group(G); group(H) |]
==> (λ〈x,y〉∈carrier(G \<Otimes> H). 〈y, x〉) ∈ G \<Otimes> H ≅ H \<Otimes> G
lemma DirProdGroup_assoc_iso:
[| group(G); group(H); group(I) |]
==> (λ〈〈x,y〉,z〉∈carrier((G \<Otimes> H) \<Otimes> I). 〈x, y, z〉) ∈
(G \<Otimes> H) \<Otimes> I ≅ G \<Otimes> H \<Otimes> I
lemma one_closed:
h ` \<one> ∈ carrier(H)
lemma hom_one:
h ` \<one> = \<one>H
lemma inv_closed:
x ∈ carrier(G) ==> h ` (inv x) ∈ carrier(H)
lemma hom_inv:
x ∈ carrier(G) ==> h ` (inv x) = invH h ` x
lemma m_lcomm:
[| x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G) |]
==> x · (y · z) = y · (x · z)
lemma m_ac:
[| x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G) |] ==> x · y · z = x · (y · z)
[| x ∈ carrier(G); y ∈ carrier(G) |] ==> x · y = y · x
[| x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G) |]
==> x · (y · z) = y · (x · z)
lemma inv_mult:
[| x ∈ carrier(G); y ∈ carrier(G) |] ==> inv (x · y) = inv x · inv y
lemma subgroup_self:
subgroup(carrier(G), G)
lemma subgroup_imp_group:
subgroup(H, G) ==> group(update_carrier(G, H))
lemma subgroupI:
[| H ⊆ carrier(G); H ≠ 0; !!a. a ∈ H ==> inv a ∈ H;
!!a b. [| a ∈ H; b ∈ H |] ==> a · b ∈ H |]
==> subgroup(H, G)
theorem group_BijGroup:
group(BijGroup(S))
lemma Bij_Inv_mem:
[| f ∈ bij(S, S); x ∈ S |] ==> converse(f) ` x ∈ S
lemma inv_BijGroup:
f ∈ bij(S, S) ==> invBijGroup(S) f = converse(f)
lemma iso_is_bij:
h ∈ G ≅ H ==> h ∈ bij(carrier(G), carrier(H))
lemma id_in_auto:
id(carrier(G)) ∈ auto(G)
lemma subgroup_auto:
subgroup(auto(G), BijGroup(carrier(G)))
theorem AutoGroup:
group(AutoGroup(G))
lemma coset_mult_assoc:
[| M ⊆ carrier(G); g ∈ carrier(G); h ∈ carrier(G) |]
==> M #> g #> h = M #> g · h
lemma coset_mult_one:
M ⊆ carrier(G) ==> M #> \<one> = M
lemma solve_equation:
[| subgroup(H, G); x ∈ H; y ∈ H |] ==> ∃h∈H. y = h · x
lemma repr_independence:
[| y ∈ H #> x; x ∈ carrier(G); subgroup(H, G) |] ==> H #> x = H #> y
lemma coset_join2:
[| x ∈ carrier(G); subgroup(H, G); x ∈ H |] ==> H #> x = H
lemma r_coset_subset_G:
[| H ⊆ carrier(G); x ∈ carrier(G) |] ==> H #> x ⊆ carrier(G)
lemma rcosI:
[| h ∈ H; H ⊆ carrier(G); x ∈ carrier(G) |] ==> h · x ∈ H #> x
lemma rcosetsI:
[| H ⊆ carrier(G); x ∈ carrier(G) |] ==> H #> x ∈ rcosets H
lemma transpose_inv:
[| x · y = z; x ∈ carrier(G); y ∈ carrier(G); z ∈ carrier(G) |]
==> inv x · z = y
lemma normal_imp_subgroup:
H \<lhd> G ==> subgroup(H, G)
lemma normalI:
[| subgroup(H, G); ∀x∈carrier(G). H #> x = x <# H |] ==> H \<lhd> G
lemma inv_op_closed1:
[| x ∈ carrier(G); h ∈ H |] ==> inv x · h · x ∈ H
lemma inv_op_closed2:
[| x ∈ carrier(G); h ∈ H |] ==> x · h · inv x ∈ H
lemma normal_inv_iff:
N \<lhd> G <-> subgroup(N, G) ∧ (∀x∈carrier(G). ∀h∈N. x · h · inv x ∈ N)
lemma l_coset_subset_G:
[| H ⊆ carrier(G); x ∈ carrier(G) |] ==> x <# H ⊆ carrier(G)
lemma l_coset_swap:
[| y ∈ x <# H; x ∈ carrier(G); subgroup(H, G) |] ==> x ∈ y <# H
lemma l_coset_carrier:
[| y ∈ x <# H; x ∈ carrier(G); subgroup(H, G) |] ==> y ∈ carrier(G)
lemma l_repr_imp_subset:
[| y ∈ x <# H; x ∈ carrier(G); subgroup(H, G) |] ==> y <# H ⊆ x <# H
lemma l_repr_independence:
[| y ∈ x <# H; x ∈ carrier(G); subgroup(H, G) |] ==> x <# H = y <# H
lemma setmult_subset_G:
[| H ⊆ carrier(G); K ⊆ carrier(G) |] ==> H <#> K ⊆ carrier(G)
lemma subgroup_mult_id:
subgroup(H, G) ==> H <#> H = H
lemma rcos_inv:
x ∈ carrier(G) ==> set_inv (H #> x) = H #> inv x
lemma setmult_rcos_assoc:
[| H ⊆ carrier(G); K ⊆ carrier(G); x ∈ carrier(G) |]
==> H <#> (K #> x) = H <#> K #> x
lemma rcos_assoc_lcos:
[| H ⊆ carrier(G); K ⊆ carrier(G); x ∈ carrier(G) |]
==> H #> x <#> K = H <#> (x <# K)
lemma rcos_mult_step1:
[| x ∈ carrier(G); y ∈ carrier(G) |]
==> H #> x <#> (H #> y) = H <#> (x <# H) #> y
lemma rcos_mult_step2:
[| x ∈ carrier(G); y ∈ carrier(G) |]
==> H <#> (x <# H) #> y = H <#> (H #> x) #> y
lemma rcos_mult_step3:
[| x ∈ carrier(G); y ∈ carrier(G) |] ==> H <#> (H #> x) #> y = H #> x · y
lemma rcos_sum:
[| x ∈ carrier(G); y ∈ carrier(G) |] ==> H #> x <#> (H #> y) = H #> x · y
lemma rcosets_mult_eq:
M ∈ rcosets H ==> H <#> M = M
lemma equiv_rcong:
group(G) ==> equiv(carrier(G), rcong H)
lemma l_coset_eq_rcong:
[| group(G); a ∈ carrier(G) |] ==> a <# H = (rcong H) `` {a}
lemma rcos_equation:
[| subgroup(H, G); ha · a = h · b; a ∈ carrier(G); b ∈ carrier(G); h ∈ H;
ha ∈ H; hb ∈ H |]
==> hb · a ∈ (\<Union>h∈H. {h · b})
lemma rcos_disjoint:
[| subgroup(H, G); a ∈ rcosets H; b ∈ rcosets H; a ≠ b |] ==> a ∩ b = 0
lemma rcos_self:
[| subgroup(H, G); x ∈ carrier(G) |] ==> x ∈ H #> x
lemma rcosets_part_G:
subgroup(H, G) ==> \<Union>(rcosets H) = carrier(G)
lemma cosets_finite:
[| c ∈ rcosets H; H ⊆ carrier(G); Finite(carrier(G)) |] ==> Finite(c)
lemma card_cosets_equal:
[| H ⊆ carrier(G); c ∈ rcosets H |] ==> |c| = |H|
lemma rcosets_subset_PowG:
subgroup(H, G) ==> rcosets H ⊆ Pow(carrier(G))
theorem lagrange:
[| Finite(carrier(G)); subgroup(H, G) |] ==> |rcosets H| #× |H| = order(G)
lemma setmult_closed:
[| K1.0 ∈ rcosets H; K2.0 ∈ rcosets H |] ==> K1.0 <#> K2.0 ∈ rcosets H
lemma setinv_closed:
K ∈ rcosets H ==> set_inv K ∈ rcosets H
lemma rcosets_assoc:
[| M1.0 ∈ rcosets H; M2.0 ∈ rcosets H; M3.0 ∈ rcosets H |]
==> M1.0 <#> M2.0 <#> M3.0 = M1.0 <#> (M2.0 <#> M3.0)
lemma subgroup_in_rcosets:
group(G) ==> H ∈ rcosets H
lemma rcosets_inv_mult_group_eq:
M ∈ rcosets H ==> set_inv M <#> M = H
theorem factorgroup_is_group:
group(G Mod H)
lemma inv_FactGroup:
X ∈ carrier(G Mod H) ==> invG Mod H X = set_inv X
lemma r_coset_hom_Mod:
(λa∈carrier(G). H #> a) ∈ hom(G, G Mod H)
lemma subgroup_kernel:
subgroup(kernel(G, H, h), G)
lemma normal_kernel:
kernel(G, H, h) \<lhd> G
lemma FactGroup_nonempty:
X ∈ carrier(G Mod kernel(G, H, h)) ==> X ≠ 0
lemma FactGroup_contents_mem:
X ∈ carrier(G Mod kernel(G, H, h)) ==> contents(h `` X) ∈ carrier(H)
lemma mult_FactGroup:
[| X ∈ carrier(G Mod H); X' ∈ carrier(G Mod H) |] ==> X ·G Mod H X' = X <#>G X'
lemma FactGroup_m_closed:
[| X ∈ carrier(G Mod H); X' ∈ carrier(G Mod H) |]
==> X <#> X' ∈ carrier(G Mod H)
lemma FactGroup_hom:
(λX∈carrier(G Mod kernel(G, H, h)). contents(h `` X)) ∈
hom(G Mod kernel(G, H, h), H)
lemma FactGroup_subset:
[| g ∈ carrier(G); g' ∈ carrier(G); h ` g = h ` g' |]
==> kernel(G, H, h) #> g ⊆ kernel(G, H, h) #> g'
lemma FactGroup_inj:
(λX∈carrier(G Mod kernel(G, H, h)). contents(h `` X)) ∈
inj(carrier(G Mod kernel(G, H, h)), carrier(H))
lemma kernel_rcoset_subset:
g ∈ carrier(G) ==> kernel(G, H, h) #> g ⊆ carrier(G)
lemma FactGroup_surj:
h ∈ surj(carrier(G), carrier(H))
==> (λX∈carrier(G Mod kernel(G, H, h)). contents(h `` X)) ∈
surj(carrier(G Mod kernel(G, H, h)), carrier(H))
theorem FactGroup_iso:
h ∈ surj(carrier(G), carrier(H))
==> (λX∈carrier(G Mod kernel(G, H, h)). contents(h `` X)) ∈
G Mod kernel(G, H, h) ≅ H