Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Classes of Conjugation. Normal Subgroups

**by****Wojciech A. Trybulec**- Received August 10, 1990
- MML identifier: GROUP_3

- [ Mizar article, MML identifier index ]

environ vocabulary REALSET1, GROUP_2, INT_1, RELAT_1, GROUP_1, BINOP_1, VECTSP_1, FUNCT_1, FINSET_1, CARD_1, ABSVALUE, BOOLE, TARSKI, RLSUB_1, SETFAM_1, ARYTM_1, GROUP_3; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XCMPLX_0, XREAL_0, NAT_1, BINOP_1, RELAT_1, FUNCT_1, FUNCT_2, STRUCT_0, RLVECT_1, FINSET_1, VECTSP_1, GROUP_2, DOMAIN_1, CARD_1, INT_1, GROUP_1; constructors WELLORD2, REAL_1, BINOP_1, GROUP_2, DOMAIN_1, NAT_1, MEMBERED, XBOOLE_0; clusters SUBSET_1, FUNCT_1, FINSET_1, GROUP_2, GROUP_1, STRUCT_0, RELSET_1, INT_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve x,y,y1,y2 for set; reserve G for Group; reserve a,b,c,d,g,h for Element of G; reserve A,B,C,D for Subset of G; reserve H,H1,H2,H3 for Subgroup of G; reserve n for Nat; reserve i for Integer; theorem :: GROUP_3:1 a * b * b" = a & a * b" * b = a & b" * b * a = a & b * b" * a = a & a * (b * b") = a & a * (b" * b) = a & b" * (b * a) = a & b * (b" * a) = a; theorem :: GROUP_3:2 G is commutative Group iff the mult of G is commutative; theorem :: GROUP_3:3 (1).G is commutative; theorem :: GROUP_3:4 A c= B & C c= D implies A * C c= B * D; theorem :: GROUP_3:5 A c= B implies a * A c= a * B & A * a c= B * a; theorem :: GROUP_3:6 H1 is Subgroup of H2 implies a * H1 c= a * H2 & H1 * a c= H2 * a; theorem :: GROUP_3:7 a * H = {a} * H; theorem :: GROUP_3:8 H * a = H * {a}; theorem :: GROUP_3:9 a * A * H = a * (A * H); theorem :: GROUP_3:10 A * a * H = A * (a * H); theorem :: GROUP_3:11 a * H * A = a * (H * A); theorem :: GROUP_3:12 A * H * a = A * (H * a); theorem :: GROUP_3:13 H * a * A = H * (a * A); theorem :: GROUP_3:14 H * A * a = H * (A * a); theorem :: GROUP_3:15 H1 * a * H2 = H1 * (a * H2); definition let G; func Subgroups G -> set means :: GROUP_3:def 1 x in it iff x is strict Subgroup of G; end; definition let G; cluster Subgroups G -> non empty; end; canceled 2; theorem :: GROUP_3:18 for G being strict Group holds G in Subgroups G; theorem :: GROUP_3:19 G is finite implies Subgroups G is finite; definition let G,a,b; func a |^ b -> Element of G equals :: GROUP_3:def 2 b" * a * b; end; theorem :: GROUP_3:20 a |^ b = b" * a * b & a |^ b = b" * (a * b); theorem :: GROUP_3:21 a |^ g = b |^ g implies a = b; theorem :: GROUP_3:22 (1.G) |^ a = 1.G; theorem :: GROUP_3:23 a |^ b = 1.G implies a = 1.G; theorem :: GROUP_3:24 a |^ 1.G = a; theorem :: GROUP_3:25 a |^ a = a; theorem :: GROUP_3:26 a |^ a" = a & a" |^ a = a"; theorem :: GROUP_3:27 a |^ b = a iff a * b = b * a; theorem :: GROUP_3:28 (a * b) |^ g = a |^ g * (b |^ g); theorem :: GROUP_3:29 a |^ g |^ h = a |^ (g * h); theorem :: GROUP_3:30 a |^ b |^ b" = a & a |^ b" |^ b = a; canceled; theorem :: GROUP_3:32 a" |^ b = (a |^ b)"; theorem :: GROUP_3:33 (a |^ n) |^ b = (a |^ b) |^ n; theorem :: GROUP_3:34 (a |^ i) |^ b = (a |^ b) |^ i; theorem :: GROUP_3:35 G is commutative Group implies a |^ b = a; theorem :: GROUP_3:36 (for a,b holds a |^ b = a) implies G is commutative; definition let G,A,B; func A |^ B -> Subset of G equals :: GROUP_3:def 3 {g |^ h : g in A & h in B}; end; canceled; theorem :: GROUP_3:38 x in A |^ B iff ex g,h st x = g |^ h & g in A & h in B; theorem :: GROUP_3:39 A |^ B <> {} iff A <> {} & B <> {}; theorem :: GROUP_3:40 A |^ B c= B" * A * B; theorem :: GROUP_3:41 (A * B) |^ C c= A |^ C * (B |^ C); theorem :: GROUP_3:42 A |^ B |^ C = A |^ (B * C); theorem :: GROUP_3:43 A" |^ B = (A |^ B)"; theorem :: GROUP_3:44 {a} |^ {b} = {a |^ b}; theorem :: GROUP_3:45 {a} |^ {b,c} = {a |^ b,a |^ c}; theorem :: GROUP_3:46 {a,b} |^ {c} = {a |^ c,b |^ c}; theorem :: GROUP_3:47 {a,b} |^ {c,d} = {a |^ c,a |^ d,b |^ c,b |^ d}; definition let G,A,g; func A |^ g -> Subset of G equals :: GROUP_3:def 4 A |^ {g}; func g |^ A -> Subset of G equals :: GROUP_3:def 5 {g} |^ A; end; canceled 2; theorem :: GROUP_3:50 x in A |^ g iff ex h st x = h |^ g & h in A; theorem :: GROUP_3:51 x in g |^ A iff ex h st x = g |^ h & h in A; theorem :: GROUP_3:52 g |^ A c= A" * g * A; theorem :: GROUP_3:53 A |^ B |^ g = A |^ (B * g); theorem :: GROUP_3:54 A |^ g |^ B = A |^ (g * B); theorem :: GROUP_3:55 g |^ A |^ B = g |^ (A * B); theorem :: GROUP_3:56 A |^ a |^ b = A |^ (a * b); theorem :: GROUP_3:57 a |^ A |^ b = a |^ (A * b); theorem :: GROUP_3:58 a |^ b |^ A = a |^ (b * A); theorem :: GROUP_3:59 A |^ g = g" * A * g; theorem :: GROUP_3:60 (A * B) |^ a c= (A |^ a) * (B |^ a); theorem :: GROUP_3:61 A |^ 1.G = A; theorem :: GROUP_3:62 A <> {} implies (1.G) |^ A = {1.G}; theorem :: GROUP_3:63 A |^ a |^ a" = A & A |^ a" |^ a = A; canceled; theorem :: GROUP_3:65 G is commutative Group iff for A,B st B <> {} holds A |^ B = A; theorem :: GROUP_3:66 G is commutative Group iff for A,g holds A |^ g = A; theorem :: GROUP_3:67 G is commutative Group iff for A,g st A <> {} holds g |^ A = {g}; definition let G,H,a; func H |^ a -> strict Subgroup of G means :: GROUP_3:def 6 the carrier of it = carr(H) |^ a; end; canceled 2; theorem :: GROUP_3:70 x in H |^ a iff ex g st x = g |^ a & g in H; theorem :: GROUP_3:71 the carrier of H |^ a = a" * H * a; theorem :: GROUP_3:72 H |^ a |^ b = H |^ (a * b); theorem :: GROUP_3:73 for H being strict Subgroup of G holds H |^ 1.G = H; theorem :: GROUP_3:74 for H being strict Subgroup of G holds H |^ a |^ a" = H & H |^ a" |^ a = H; canceled; theorem :: GROUP_3:76 (H1 /\ H2) |^ a = (H1 |^ a) /\ (H2 |^ a); theorem :: GROUP_3:77 Ord H = Ord(H |^ a); theorem :: GROUP_3:78 H is finite iff H |^ a is finite; theorem :: GROUP_3:79 H is finite implies ord H = ord(H |^ a); theorem :: GROUP_3:80 (1).G |^ a = (1).G; theorem :: GROUP_3:81 for H being strict Subgroup of G holds H |^ a = (1).G implies H = (1).G; theorem :: GROUP_3:82 for G being Group, a being Element of G holds (Omega).G |^ a = (Omega).G; theorem :: GROUP_3:83 for H being strict Subgroup of G holds H |^ a = G implies H = G; theorem :: GROUP_3:84 Index H = Index(H |^ a); theorem :: GROUP_3:85 Left_Cosets H is finite implies index H = index(H |^ a); theorem :: GROUP_3:86 G is commutative Group implies for H being strict Subgroup of G for a holds H |^ a = H; definition let G,a,b; pred a,b are_conjugated means :: GROUP_3:def 7 ex g st a = b |^ g; antonym a,b are_not_conjugated; end; canceled; theorem :: GROUP_3:88 a,b are_conjugated iff ex g st b = a |^ g; theorem :: GROUP_3:89 a,a are_conjugated; theorem :: GROUP_3:90 a,b are_conjugated implies b,a are_conjugated; definition let G,a,b; redefine pred a,b are_conjugated; reflexivity; symmetry; end; theorem :: GROUP_3:91 a,b are_conjugated & b,c are_conjugated implies a,c are_conjugated; theorem :: GROUP_3:92 a,1.G are_conjugated or 1.G,a are_conjugated implies a = 1.G; theorem :: GROUP_3:93 a |^ carr (Omega).G = {b : a,b are_conjugated}; definition let G,a; func con_class a -> Subset of G equals :: GROUP_3:def 8 a |^ carr (Omega).G; end; canceled; theorem :: GROUP_3:95 x in con_class a iff ex b st b = x & a,b are_conjugated; theorem :: GROUP_3:96 a in con_class b iff a,b are_conjugated; theorem :: GROUP_3:97 a |^ g in con_class a; theorem :: GROUP_3:98 a in con_class a; theorem :: GROUP_3:99 a in con_class b implies b in con_class a; theorem :: GROUP_3:100 con_class a = con_class b iff con_class a meets con_class b; theorem :: GROUP_3:101 con_class a = {1.G} iff a = 1.G; theorem :: GROUP_3:102 con_class a * A = A * con_class a; definition let G,A,B; pred A,B are_conjugated means :: GROUP_3:def 9 ex g st A = B |^ g; antonym A,B are_not_conjugated; end; canceled; theorem :: GROUP_3:104 A,B are_conjugated iff ex g st B = A |^ g; theorem :: GROUP_3:105 A,A are_conjugated; theorem :: GROUP_3:106 A,B are_conjugated implies B,A are_conjugated; definition let G,A,B; redefine pred A,B are_conjugated; reflexivity; symmetry; end; theorem :: GROUP_3:107 A,B are_conjugated & B,C are_conjugated implies A,C are_conjugated; theorem :: GROUP_3:108 {a},{b} are_conjugated iff a,b are_conjugated; theorem :: GROUP_3:109 A,carr H1 are_conjugated implies ex H2 being strict Subgroup of G st the carrier of H2 = A; definition let G,A; func con_class A -> Subset-Family of G equals :: GROUP_3:def 10 {B : A,B are_conjugated}; end; canceled; theorem :: GROUP_3:111 x in con_class A iff ex B st x = B & A,B are_conjugated; canceled; theorem :: GROUP_3:113 A in con_class B iff A,B are_conjugated; theorem :: GROUP_3:114 A |^ g in con_class A; theorem :: GROUP_3:115 A in con_class A; theorem :: GROUP_3:116 A in con_class B implies B in con_class A; theorem :: GROUP_3:117 con_class A = con_class B iff con_class A meets con_class B; theorem :: GROUP_3:118 con_class{a} = {{b} : b in con_class a}; theorem :: GROUP_3:119 G is finite implies con_class A is finite; definition let G,H1,H2; pred H1,H2 are_conjugated means :: GROUP_3:def 11 ex g st the HGrStr of H1 = H2 |^ g; antonym H1,H2 are_not_conjugated; end; canceled; theorem :: GROUP_3:121 for H1,H2 being strict Subgroup of G holds H1,H2 are_conjugated iff ex g st H2 = H1 |^ g; theorem :: GROUP_3:122 for H1 being strict Subgroup of G holds H1,H1 are_conjugated; theorem :: GROUP_3:123 for H1,H2 being strict Subgroup of G holds H1,H2 are_conjugated implies H2,H1 are_conjugated; definition let G; let H1,H2 be strict Subgroup of G; redefine pred H1,H2 are_conjugated; reflexivity; symmetry; end; theorem :: GROUP_3:124 for H1,H2 being strict Subgroup of G holds H1,H2 are_conjugated & H2,H3 are_conjugated implies H1,H3 are_conjugated; reserve L for Subset of Subgroups G; definition let G,H; func con_class H -> Subset of Subgroups G means :: GROUP_3:def 12 x in it iff ex H1 being strict Subgroup of G st x = H1 & H,H1 are_conjugated; end; canceled 2; theorem :: GROUP_3:127 x in con_class H implies x is strict Subgroup of G; theorem :: GROUP_3:128 for H1,H2 being strict Subgroup of G holds H1 in con_class H2 iff H1,H2 are_conjugated; theorem :: GROUP_3:129 for H being strict Subgroup of G holds H |^ g in con_class H; theorem :: GROUP_3:130 for H being strict Subgroup of G holds H in con_class H; theorem :: GROUP_3:131 for H1,H2 being strict Subgroup of G holds H1 in con_class H2 implies H2 in con_class H1; theorem :: GROUP_3:132 for H1,H2 being strict Subgroup of G holds con_class H1 = con_class H2 iff con_class H1 meets con_class H2; theorem :: GROUP_3:133 G is finite implies con_class H is finite; theorem :: GROUP_3:134 for H1 being strict Subgroup of G holds H1,H2 are_conjugated iff carr H1,carr H2 are_conjugated; definition let G; let IT be Subgroup of G; attr IT is normal means :: GROUP_3:def 13 for a holds IT |^ a = the HGrStr of IT; end; definition let G; cluster strict normal Subgroup of G; end; reserve N2 for normal Subgroup of G; canceled 2; theorem :: GROUP_3:137 (1).G is normal & (Omega).G is normal; theorem :: GROUP_3:138 for N1,N2 being strict normal Subgroup of G holds N1 /\ N2 is normal; theorem :: GROUP_3:139 for H being strict Subgroup of G holds G is commutative Group implies H is normal; theorem :: GROUP_3:140 H is normal Subgroup of G iff for a holds a * H = H * a; theorem :: GROUP_3:141 for H being Subgroup of G holds H is normal Subgroup of G iff for a holds a * H c= H * a; theorem :: GROUP_3:142 for H being Subgroup of G holds H is normal Subgroup of G iff for a holds H * a c= a * H; theorem :: GROUP_3:143 for H being Subgroup of G holds H is normal Subgroup of G iff for A holds A * H = H * A; theorem :: GROUP_3:144 for H being strict Subgroup of G holds H is normal Subgroup of G iff for a holds H is Subgroup of H |^ a; theorem :: GROUP_3:145 for H being strict Subgroup of G holds H is normal Subgroup of G iff for a holds H |^ a is Subgroup of H; theorem :: GROUP_3:146 for H being strict Subgroup of G holds H is normal Subgroup of G iff con_class H = {H}; theorem :: GROUP_3:147 for H being strict Subgroup of G holds H is normal Subgroup of G iff for a st a in H holds con_class a c= carr H; theorem :: GROUP_3:148 for N1,N2 being strict normal Subgroup of G holds carr N1 * carr N2 = carr N2 * carr N1; theorem :: GROUP_3:149 for N1,N2 being strict normal Subgroup of G ex N being strict normal Subgroup of G st the carrier of N = carr N1 * carr N2; theorem :: GROUP_3:150 for N being normal Subgroup of G holds Left_Cosets N = Right_Cosets N; theorem :: GROUP_3:151 for H being Subgroup of G holds Left_Cosets H is finite & index H = 2 implies H is normal Subgroup of G; definition let G; let A; func Normalizator A -> strict Subgroup of G means :: GROUP_3:def 14 the carrier of it = {h : A |^ h = A}; end; canceled 2; theorem :: GROUP_3:154 x in Normalizator A iff ex h st x = h & A |^ h = A; theorem :: GROUP_3:155 Card con_class A = Index Normalizator A; theorem :: GROUP_3:156 con_class A is finite or Left_Cosets Normalizator A is finite implies ex C being finite set st C = con_class A & card C = index Normalizator A; theorem :: GROUP_3:157 Card con_class a = Index Normalizator{a}; theorem :: GROUP_3:158 con_class a is finite or Left_Cosets Normalizator{a} is finite implies ex C being finite set st C = con_class a & card C = index Normalizator{a}; definition let G; let H; func Normalizator H -> strict Subgroup of G equals :: GROUP_3:def 15 Normalizator carr H; end; canceled; theorem :: GROUP_3:160 for H being strict Subgroup of G holds x in Normalizator H iff ex h st x = h & H |^ h = H; theorem :: GROUP_3:161 for H being strict Subgroup of G holds Card con_class H = Index Normalizator H; theorem :: GROUP_3:162 for H being strict Subgroup of G holds con_class H is finite or Left_Cosets Normalizator H is finite implies ex C being finite set st C = con_class H & card C = index Normalizator H; theorem :: GROUP_3:163 for G being strict Group, H being strict Subgroup of G holds H is normal Subgroup of G iff Normalizator H = G; theorem :: GROUP_3:164 for G being strict Group holds Normalizator (1).G = G; theorem :: GROUP_3:165 for G being strict Group holds Normalizator (Omega).G = G; :: :: Auxiliary theorems. :: theorem :: GROUP_3:166 for X being finite set st card X = 2 ex x,y st x <> y & X = {x,y};

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