Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Homomorphisms and Isomorphisms of Groups. Quotient Group

**by****Wojciech A. Trybulec, and****Michal J. Trybulec**- Received October 3, 1991
- MML identifier: GROUP_6

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, REALSET1, GROUP_2, GROUP_3, RLSUB_1, INT_1, BOOLE, RELAT_1, GROUP_1, RCOMP_1, SUBSET_1, GROUP_4, GROUP_5, VECTSP_1, NATTRA_1, FINSET_1, CARD_1, BINOP_1, QC_LANG1, ABSVALUE, WELLORD1, LATTICES, GROUP_6; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XREAL_0, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, STRUCT_0, CARD_1, FINSET_1, BINOP_1, REALSET1, INT_1, RLVECT_1, VECTSP_1, GROUP_1, GROUP_2, GROUP_3, NAT_1, GROUP_4, GROUP_5; constructors DOMAIN_1, BINOP_1, REALSET1, NAT_1, GROUP_4, GROUP_5, MEMBERED, PARTFUN1, RELAT_2, XBOOLE_0; clusters FUNCT_1, FINSET_1, GROUP_1, GROUP_2, GROUP_3, STRUCT_0, RELSET_1, INT_1, FUNCT_2, MEMBERED, ZFMISC_1, PARTFUN1, XBOOLE_0; requirements NUMERALS, SUBSET, BOOLE; begin theorem :: GROUP_6:1 for A,B being non empty set, f being Function of A,B holds f is one-to-one iff for a,b being Element of A st f.a = f.b holds a = b; definition let G be Group, A be Subgroup of G; redefine mode Subgroup of A -> Subgroup of G; end; definition let G be Group; cluster (1).G -> normal; cluster (Omega).G -> normal; end; reserve n for Nat; reserve i for Integer; reserve G,H,I for Group; reserve A,B for Subgroup of G; reserve N for normal Subgroup of G; reserve a,a1,a2,a3,b,b1 for Element of G; reserve c,d for Element of H; reserve f for Function of the carrier of G, the carrier of H; reserve x,y,y1,y2,z for set; reserve A1,A2 for Subset of G; theorem :: GROUP_6:2 for X being Subgroup of A, x being Element of A st x = a holds x * X = a * (X qua Subgroup of G) & X * x = (X qua Subgroup of G) * a; theorem :: GROUP_6:3 for X,Y being Subgroup of A holds (X qua Subgroup of G) /\ (Y qua Subgroup of G) = X /\ Y; theorem :: GROUP_6:4 a * b * a" = b |^ a" & a * (b * a") = b |^ a"; canceled; theorem :: GROUP_6:6 a * A * A = a * A & a * (A * A) = a * A & A * A * a = A * a & A * (A * a) = A *a; theorem :: GROUP_6:7 for G being Group, A1 being Subset of G holds A1 = {[.a,b.] where a is Element of G, b is Element of G : not contradiction} implies G` = gr A1; theorem :: GROUP_6:8 for G being strict Group, B being strict Subgroup of G holds G` is Subgroup of B iff for a,b being Element of G holds [.a,b.] in B; theorem :: GROUP_6:9 for N being normal Subgroup of G holds N is Subgroup of B implies N is normal Subgroup of B; definition let G,B; let M be normal Subgroup of G; assume the HGrStr of M is Subgroup of B; func (B,M)`*` -> strict normal Subgroup of B equals :: GROUP_6:def 1 the HGrStr of M; end; theorem :: GROUP_6:10 B /\ N is normal Subgroup of B & N /\ B is normal Subgroup of B; definition let G,B; let N be normal Subgroup of G; redefine func B /\ N -> strict normal Subgroup of B; end; definition let G; let N be normal Subgroup of G; let B; redefine func N /\ B -> strict normal Subgroup of B; end; definition let G be non empty 1-sorted; redefine attr G is trivial means :: GROUP_6:def 2 ex x st the carrier of G = {x}; end; definition cluster trivial Group; end; theorem :: GROUP_6:11 (1).G is trivial; theorem :: GROUP_6:12 G is trivial iff ord G = 1 & G is finite; theorem :: GROUP_6:13 for G being strict Group holds G is trivial implies (1).G = G; definition let G,N; func Cosets N -> set equals :: GROUP_6:def 3 Left_Cosets N; end; definition let G,N; cluster Cosets N -> non empty; end; theorem :: GROUP_6:14 for N being normal Subgroup of G holds Cosets N = Left_Cosets N & Cosets N = Right_Cosets N; theorem :: GROUP_6:15 for N being normal Subgroup of G holds x in Cosets N implies ex a st x = a * N & x = N * a; theorem :: GROUP_6:16 for N being normal Subgroup of G holds a * N in Cosets N & N * a in Cosets N; theorem :: GROUP_6:17 for N being normal Subgroup of G holds x in Cosets N implies x is Subset of G; theorem :: GROUP_6:18 for N being normal Subgroup of G holds A1 in Cosets N & A2 in Cosets N implies A1 * A2 in Cosets N; definition let G; let N be normal Subgroup of G; func CosOp N -> BinOp of Cosets N means :: GROUP_6:def 4 for W1,W2 being Element of Cosets N for A1,A2 st W1 = A1 & W2 = A2 holds it.(W1,W2) = A1 * A2; end; definition let G; let N be normal Subgroup of G; func G./.N -> HGrStr equals :: GROUP_6:def 5 HGrStr (# Cosets N, CosOp N #); end; definition let G; let N be normal Subgroup of G; cluster G./.N -> strict non empty; end; canceled 3; theorem :: GROUP_6:22 for N being normal Subgroup of G holds the carrier of G./.N = Cosets N; theorem :: GROUP_6:23 for N being normal Subgroup of G holds the mult of G./.N = CosOp N; reserve N for normal Subgroup of G; reserve S,T1,T2 for Element of G./.N; definition let G,N,S; func @S -> Subset of G equals :: GROUP_6:def 6 S; end; theorem :: GROUP_6:24 for N being normal Subgroup of G, T1,T2 being Element of G./.N holds @T1 * @T2 = T1 * T2; theorem :: GROUP_6:25 @(T1 * T2) = @T1 * @T2; definition let G; let N be normal Subgroup of G; cluster G./.N -> associative Group-like; end; theorem :: GROUP_6:26 for N being normal Subgroup of G, S being Element of G./.N ex a st S = a * N & S = N * a; theorem :: GROUP_6:27 N * a is Element of G./.N & a * N is Element of G./.N & carr N is Element of G./.N; theorem :: GROUP_6:28 for N being normal Subgroup of G holds x in G./.N iff ex a st x = a * N & x = N * a; theorem :: GROUP_6:29 for N being normal Subgroup of G holds 1.(G./.N) = carr N; theorem :: GROUP_6:30 for N being normal Subgroup of G, S being Element of G./.N holds S = a * N implies S" = a" * N; theorem :: GROUP_6:31 for N being normal Subgroup of G holds Left_Cosets N is finite implies G./.N is finite; theorem :: GROUP_6:32 for N being normal Subgroup of G holds Ord(G./.N) = Index N; theorem :: GROUP_6:33 for N being normal Subgroup of G holds Left_Cosets N is finite implies ord(G./.N) = index N; theorem :: GROUP_6:34 for M being strict normal Subgroup of G holds M is Subgroup of B implies B./.(B,M)`*` is Subgroup of G./.M; theorem :: GROUP_6:35 for N,M being strict normal Subgroup of G holds M is Subgroup of N implies N./.(N,M)`*` is normal Subgroup of G./.M; theorem :: GROUP_6:36 for G being strict Group, N be strict normal Subgroup of G holds G./.N is commutative Group iff G` is Subgroup of N; definition let G,H; mode Homomorphism of G,H -> Function of the carrier of G, the carrier of H means :: GROUP_6:def 7 it.(a * b) = it.a * it.b; end; reserve g,h for Homomorphism of G,H; reserve g1 for Homomorphism of H,G; reserve h1 for Homomorphism of H,I; canceled 3; theorem :: GROUP_6:40 g.(1.G) = 1.H; theorem :: GROUP_6:41 g.(a") = (g.a)"; theorem :: GROUP_6:42 g.(a |^ b) = (g.a) |^ (g.b); theorem :: GROUP_6:43 g. [.a,b.] = [.g.a,g.b.]; theorem :: GROUP_6:44 g. [.a1,a2,a3.] = [.g.a1,g.a2,g.a3.]; theorem :: GROUP_6:45 g.(a |^ n) = (g.a) |^ n; theorem :: GROUP_6:46 g.(a |^ i) = (g.a) |^ i; theorem :: GROUP_6:47 id the carrier of G is Homomorphism of G,G; theorem :: GROUP_6:48 h1 * h is Homomorphism of G,I; definition let G,H,I,h,h1; redefine func h1 * h -> Homomorphism of G,I; end; definition let G,H,g; redefine func rng g -> Subset of H; end; definition let G,H; func 1:(G,H) -> Homomorphism of G,H means :: GROUP_6:def 8 for a holds it.a = 1.H; end; theorem :: GROUP_6:49 h1 * 1:(G,H) = 1:(G,I) & 1:(H,I) * h = 1:(G,I); definition let G; let N be normal Subgroup of G; func nat_hom N -> Homomorphism of G,G./.N means :: GROUP_6:def 9 for a holds it.a = a * N; end; definition let G,H,g; func Ker g -> strict Subgroup of G means :: GROUP_6:def 10 the carrier of it = {a : g.a = 1.H}; end; definition let G,H,g; cluster Ker g -> normal; end; theorem :: GROUP_6:50 a in Ker h iff h.a = 1.H; theorem :: GROUP_6:51 for G,H being strict Group holds Ker 1:(G,H) = G; theorem :: GROUP_6:52 for N being strict normal Subgroup of G holds Ker nat_hom N = N; definition let G,H,g; func Image g -> strict Subgroup of H means :: GROUP_6:def 11 the carrier of it = g .: (the carrier of G); end; theorem :: GROUP_6:53 rng g = the carrier of Image g; theorem :: GROUP_6:54 x in Image g iff ex a st x = g.a; theorem :: GROUP_6:55 Image g = gr rng g; theorem :: GROUP_6:56 Image 1:(G,H) = (1).H; theorem :: GROUP_6:57 for N being normal Subgroup of G holds Image nat_hom N = G./.N; theorem :: GROUP_6:58 h is Homomorphism of G,Image h; theorem :: GROUP_6:59 G is finite implies Image g is finite; theorem :: GROUP_6:60 G is commutative Group implies Image g is commutative; theorem :: GROUP_6:61 Ord Image g <=` Ord G; theorem :: GROUP_6:62 G is finite implies ord Image g <= ord G; definition let G,H,h; attr h is being_monomorphism means :: GROUP_6:def 12 h is one-to-one; synonym h is_monomorphism; attr h is being_epimorphism means :: GROUP_6:def 13 rng h = the carrier of H; synonym h is_epimorphism; end; theorem :: GROUP_6:63 h is_monomorphism & c in Image h implies h.(h".c) = c; theorem :: GROUP_6:64 h is_monomorphism implies h".(h.a) = a; theorem :: GROUP_6:65 h is_monomorphism implies h" is Homomorphism of Image h,G; theorem :: GROUP_6:66 h is_monomorphism iff Ker h = (1).G; theorem :: GROUP_6:67 for H being strict Group, h being Homomorphism of G,H holds h is_epimorphism iff Image h = H; theorem :: GROUP_6:68 for H being strict Group, h being Homomorphism of G,H holds h is_epimorphism implies for c being Element of H ex a st h.a = c; theorem :: GROUP_6:69 for N being normal Subgroup of G holds nat_hom N is_epimorphism; definition let G,H,h; attr h is being_isomorphism means :: GROUP_6:def 14 h is_epimorphism & h is_monomorphism; synonym h is_isomorphism; end; theorem :: GROUP_6:70 h is_isomorphism iff rng h = the carrier of H & h is one-to-one; theorem :: GROUP_6:71 h is_isomorphism implies dom h = the carrier of G & rng h = the carrier of H ; theorem :: GROUP_6:72 for H being strict Group, h being Homomorphism of G,H holds h is_isomorphism implies h" is Homomorphism of H,G; theorem :: GROUP_6:73 h is_isomorphism & g1 = h" implies g1 is_isomorphism; theorem :: GROUP_6:74 h is_isomorphism & h1 is_isomorphism implies h1 * h is_isomorphism; theorem :: GROUP_6:75 for G being Group holds nat_hom (1).G is_isomorphism; definition let G,H; pred G,H are_isomorphic means :: GROUP_6:def 15 ex h st h is_isomorphism; reflexivity; end; canceled; theorem :: GROUP_6:77 for G,H being strict Group holds G,H are_isomorphic implies H,G are_isomorphic; theorem :: GROUP_6:78 G,H are_isomorphic & H,I are_isomorphic implies G,I are_isomorphic; theorem :: GROUP_6:79 h is_monomorphism implies G,Image h are_isomorphic; theorem :: GROUP_6:80 for G,H being strict Group holds G is trivial & H is trivial implies G,H are_isomorphic; theorem :: GROUP_6:81 (1).G,(1).H are_isomorphic; theorem :: GROUP_6:82 for G being strict Group holds G,G./.(1).G are_isomorphic & G./.(1).G,G are_isomorphic; theorem :: GROUP_6:83 for G being Group holds G./.(Omega).G is trivial; theorem :: GROUP_6:84 for G,H being strict Group, h being Homomorphism of G,H holds G,H are_isomorphic implies Ord G = Ord H; theorem :: GROUP_6:85 for G,H being strict Group holds G,H are_isomorphic & (G is finite or H is finite) implies (G is finite & H is finite); theorem :: GROUP_6:86 for G,H being strict Group holds G,H are_isomorphic & (G is finite or H is finite) implies ord G = ord H; theorem :: GROUP_6:87 for G,H being strict Group holds G,H are_isomorphic & G is trivial implies H is trivial; theorem :: GROUP_6:88 for G,H being strict Group holds G,H are_isomorphic & (G is trivial or H is trivial) implies G is trivial & H is trivial; theorem :: GROUP_6:89 for G,H being strict Group, h being Homomorphism of G,H holds G,H are_isomorphic & (G is commutative Group or H is commutative Group) implies (G is commutative Group & H is commutative Group); theorem :: GROUP_6:90 G./.Ker g,Image g are_isomorphic & Image g, G./.Ker g are_isomorphic; theorem :: GROUP_6:91 ex h being Homomorphism of G./.Ker g,Image g st h is_isomorphism & g = h * nat_hom Ker g; theorem :: GROUP_6:92 for M being strict normal Subgroup of G for J being strict normal Subgroup of G./.M st J = N./.(N,M)`*` & M is Subgroup of N holds (G./.M)./.J,G./.N are_isomorphic; theorem :: GROUP_6:93 for N being strict normal Subgroup of G holds (B "\/" N)./.(B "\/" N,N)`*`, B./.(B /\ N) are_isomorphic;

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