:: Homomorphisms and Isomorphisms of Groups. Quotient Group
:: by Wojciech A. Trybulec and Micha{\l} J. Trybulec
::
:: Received October 3, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies XBOOLE_0, FUNCT_1, SUBSET_1, GROUP_1, GROUP_2, PRE_TOPC, RLSUB_1,
NUMBERS, INT_1, STRUCT_0, RELAT_1, TARSKI, NEWTON, XXREAL_1, GROUP_4,
GROUP_5, ALGSTR_0, NATTRA_1, ZFMISC_1, CARD_1, FINSET_1, BINOP_1,
QC_LANG1, CQC_SIM1, REALSET1, MSSUBFAM, FUNCOP_1, ARYTM_3, XXREAL_0,
COMPLEX1, FUNCT_2, WELLORD1, EQREL_1, GROUP_6, NAT_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, REALSET1, CARD_1, ORDINAL1,
NUMBERS, XXREAL_0, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2,
FUNCOP_1, FINSET_1, BINOP_1, INT_1, STRUCT_0, ALGSTR_0, GROUP_1, GROUP_2,
GROUP_3, NAT_1, GROUP_4, GROUP_5, INT_2;
constructors RELAT_2, PARTFUN1, BINOP_1, FUNCOP_1, FINSUB_1, XXREAL_0, NAT_1,
REAL_1, INT_2, REALSET2, GROUP_4, GROUP_5, RELSET_1, BINOP_2;
registrations FUNCT_1, PARTFUN1, FUNCT_2, FINSET_1, XREAL_0, INT_1, STRUCT_0,
GROUP_1, GROUP_2, GROUP_3, ORDINAL1, RELSET_1;
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;
registration
let G be Group;
cluster (1).G -> normal;
cluster (Omega).G -> normal;
end;
reserve n for Element of 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";
theorem :: GROUP_6:5
a * A * A = a * A & a * (A * A) = a * A & A * A * a = A * a & A *
(A * a) = A *a;
theorem :: GROUP_6:6
for G being Group, A1 being Subset of G holds A1 = the set of all
[.a,b.] where
a is Element of G, b is Element of G implies G` = gr A1;
theorem :: GROUP_6:7
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_6:8
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 multMagma of M is Subgroup of B;
func (B,M)`*` -> strict normal Subgroup of B equals
:: GROUP_6:def 1
the multMagma of
M;
end;
theorem :: GROUP_6:9
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 being object st the carrier of G = {x};
end;
theorem :: GROUP_6:10
(1).G is trivial;
registration
let G;
cluster (1).G -> trivial;
end;
registration
cluster strict trivial for Group;
end;
theorem :: GROUP_6:11
(for G being trivial Group holds card G = 1 & G is finite) & for
G being finite Group st card G = 1 holds G is trivial;
theorem :: GROUP_6:12
for G being strict trivial Group holds (1).G = G;
notation
let G,N;
synonym Cosets N for Left_Cosets N;
end;
registration
let G,N;
cluster Cosets N -> non empty;
end;
theorem :: GROUP_6:13
for x being object holds
for N being normal Subgroup of G holds x in Cosets N implies ex
a st x = a * N & x = N * a;
theorem :: GROUP_6:14
for N being normal Subgroup of G holds a * N in Cosets N & N * a in Cosets N;
theorem :: GROUP_6:15
for N being normal Subgroup of G holds x in Cosets N implies x
is Subset of G;
theorem :: GROUP_6:16
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 3
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 -> multMagma equals
:: GROUP_6:def 4
multMagma (# Cosets N, CosOp N #);
end;
registration
let G;
let N be normal Subgroup of G;
cluster G./.N -> strict non empty;
end;
theorem :: GROUP_6:17
for N being normal Subgroup of G holds the carrier of G./.N = Cosets N;
theorem :: GROUP_6:18
for N being normal Subgroup of G holds the multF 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 5
S;
end;
theorem :: GROUP_6:19
for N being normal Subgroup of G, T1,T2 being Element of G./.N holds @
T1 * @T2 = T1 * T2;
theorem :: GROUP_6:20
@(T1 * T2) = @T1 * @T2;
registration
let G;
let N be normal Subgroup of G;
cluster G./.N -> associative Group-like;
end;
theorem :: GROUP_6:21
for N being normal Subgroup of G, S being Element of G./.N
ex a st S = a * N & S = N * a;
theorem :: GROUP_6:22
N * a is Element of G./.N & a * N is Element of G./.N &
carr N is Element of G./.N;
theorem :: GROUP_6:23
for x being object holds
for N being normal Subgroup of G holds x in G./.N iff
ex a st x = a * N & x = N * a;
theorem :: GROUP_6:24
for N being normal Subgroup of G holds 1_(G./.N) = carr N;
theorem :: GROUP_6:25
for N being normal Subgroup of G, S being Element of G./.N st S
= a * N holds S" = a" * N;
theorem :: GROUP_6:26
for N being normal Subgroup of G holds card(G./.N) = Index N;
theorem :: GROUP_6:27
for N being normal Subgroup of G holds Left_Cosets N is finite implies
card(G./.N) = index N;
theorem :: GROUP_6:28
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_6:29
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_6:30
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 be non empty multMagma;
let f be Function of G, H;
attr f is multiplicative means
:: GROUP_6:def 6
for a, b being Element of G holds f.(a * b) = f.a * f.b;
end;
registration
let G, H be unital non empty multMagma;
cluster multiplicative for Function of G, H;
end;
definition
let G,H;
mode Homomorphism of G,H is multiplicative Function of G, H;
end;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;
theorem :: GROUP_6:31
g.(1_G) = 1_H;
registration
let G,H;
cluster -> unity-preserving for Homomorphism of G,H;
end;
theorem :: GROUP_6:32
g.(a") = (g.a)";
theorem :: GROUP_6:33
g.(a |^ b) = (g.a) |^ (g.b);
theorem :: GROUP_6:34
g. [.a,b.] = [.g.a,g.b.];
theorem :: GROUP_6:35
g. [.a1,a2,a3.] = [.g.a1,g.a2,g.a3.];
theorem :: GROUP_6:36
g.(a |^ n) = (g.a) |^ n;
theorem :: GROUP_6:37
g.(a |^ i) = (g.a) |^ i;
theorem :: GROUP_6:38
for G being non empty multMagma holds
id the carrier of G is multiplicative;
theorem :: GROUP_6:39
for G,H,I being unital non empty multMagma,
h being multiplicative Function of G,H,
h1 being multiplicative Function of H,I holds
h1 * h is multiplicative;
registration
let G,H,I be unital non empty multMagma,
h be multiplicative Function of G,H,
h1 be multiplicative Function of H,I;
cluster h1 * h -> multiplicative for Function of G,I;
end;
definition
let G,H be non empty multMagma;
func 1:(G,H) -> Function of G,H equals
:: GROUP_6:def 7
G --> 1_H;
end;
registration
let G,H be unital non empty multMagma;
cluster 1:(G,H) -> multiplicative;
end;
theorem :: GROUP_6:40
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 -> Function of G,G./.N means
:: GROUP_6:def 8
for a holds it.a = a * N;
end;
registration
let G;
let N be normal Subgroup of G;
cluster nat_hom N -> multiplicative;
end;
definition
let G,H,g;
func Ker g -> strict Subgroup of G means
:: GROUP_6:def 9
the carrier of it = {a : g. a = 1_H};
end;
registration
let G,H,g;
cluster Ker g -> normal;
end;
theorem :: GROUP_6:41
a in Ker h iff h.a = 1_H;
theorem :: GROUP_6:42
for G,H being strict Group holds Ker 1:(G,H) = G;
theorem :: GROUP_6:43
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 10
the carrier of it = g .: (the carrier of G);
end;
theorem :: GROUP_6:44
rng g = the carrier of Image g;
theorem :: GROUP_6:45
for x being object holds x in Image g iff ex a st x = g.a;
theorem :: GROUP_6:46
Image g = gr rng g;
theorem :: GROUP_6:47
Image 1:(G,H) = (1).H;
theorem :: GROUP_6:48
for N being normal Subgroup of G holds Image nat_hom N = G./.N;
theorem :: GROUP_6:49
h is Homomorphism of G,Image h;
theorem :: GROUP_6:50
G is finite implies Image g is finite;
registration
let G be finite Group;
let H be Group;
let g be Homomorphism of G,H;
cluster Image g -> finite;
end;
theorem :: GROUP_6:51
G is commutative Group implies Image g is commutative;
theorem :: GROUP_6:52
card Image g c= card G;
theorem :: GROUP_6:53
for G being finite Group, H being Group, g being Homomorphism of G,H
holds card Image g <= card G;
theorem :: GROUP_6:54
h is one-to-one & c in Image h implies h.(h".c) = c;
theorem :: GROUP_6:55
h is one-to-one implies h" is Homomorphism of Image h,G;
theorem :: GROUP_6:56
h is one-to-one iff Ker h = (1).G;
theorem :: GROUP_6:57
for H being strict Group, h being Homomorphism of G,H holds
h is onto iff Image h = H;
theorem :: GROUP_6:58
for G,H being non empty set, h being Function of G,H st
h is onto for c being Element of H ex a being Element of G st h.a = c;
theorem :: GROUP_6:59
for N being normal Subgroup of G holds nat_hom N is onto;
theorem :: GROUP_6:60
for G,H being set for h being Function of G,H holds
h is bijective iff rng h = H & h is one-to-one;
theorem :: GROUP_6:61
for G being set, H being non empty set for h being Function of G,H holds
h is bijective implies dom h = G & rng h = H;
theorem :: GROUP_6:62
for H being Group, h being Homomorphism of G,H st
h is bijective holds h" is Homomorphism of H,G;
theorem :: GROUP_6:63
for G being set, H being non empty set for h being Function of G,H holds
for g1 being Function of H,G holds
h is bijective & g1 = h" implies g1 is bijective;
theorem :: GROUP_6:64
for G being set, H,I being non empty set for h being Function of G,H holds
for h1 being Function of H,I holds
h is bijective & h1 is bijective implies h1 * h is bijective;
theorem :: GROUP_6:65
for G being Group holds nat_hom (1).G is bijective;
definition
let G,H;
pred G,H are_isomorphic means
:: GROUP_6:def 11
ex h st h is bijective;
reflexivity;
end;
theorem :: GROUP_6:66
for G,H being Group holds G,H are_isomorphic implies
H,G are_isomorphic;
definition
let G,H be Group;
redefine pred G,H are_isomorphic;
symmetry;
end;
theorem :: GROUP_6:67
G,H are_isomorphic & H,I are_isomorphic implies G,I are_isomorphic;
theorem :: GROUP_6:68
h is one-to-one implies G,Image h are_isomorphic;
theorem :: GROUP_6:69
for G,H being strict Group holds G is trivial & H is trivial
implies G,H are_isomorphic;
theorem :: GROUP_6:70
(1).G,(1).H are_isomorphic;
theorem :: GROUP_6:71
for G being Group holds G,G./.(1).G are_isomorphic;
theorem :: GROUP_6:72
for G being Group holds G./.(Omega).G is trivial;
theorem :: GROUP_6:73
for G,H being strict Group holds G,H are_isomorphic implies card G = card H;
theorem :: GROUP_6:74
G,H are_isomorphic & G is finite implies H is finite;
theorem :: GROUP_6:75
for G,H being strict Group holds G,H are_isomorphic implies card G = card H;
theorem :: GROUP_6:76
for G being strict trivial Group, H being strict Group holds
G,H are_isomorphic implies H is trivial;
theorem :: GROUP_6:77
for H being Group st G,H are_isomorphic & G is commutative
holds H is commutative;
theorem :: GROUP_6:78
G./.Ker g, Image g are_isomorphic;
::$N First isomorphism theorem for groups
theorem :: GROUP_6:79
ex h being Homomorphism of G./.Ker g,Image g st h is bijective & g = h
* nat_hom Ker g;
::$N Third isomorphism theorem for groups
theorem :: GROUP_6:80
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;
::$N Second isomorphism theorem for groups
theorem :: GROUP_6:81
for N being strict normal Subgroup of G holds (B "\/" N)./.(B "\/" N,N
)`*`, B./.(B /\ N) are_isomorphic;