Journal of Formalized Mathematics
Volume 6, 1994
University of Bialystok
Copyright (c) 1994 Association of Mizar Users

### Solvable Groups

by

MML identifier: GRSOLV_1
[ Mizar article, MML identifier index ]

```environ

vocabulary REALSET1, FINSEQ_1, GROUP_3, FUNCT_1, RLSUB_1, GROUP_2, RELAT_1,
GROUP_6, BINOP_1, BOOLE, QC_LANG1, WELLORD1, GROUP_1, VECTSP_1, GRAPH_1,
GRSOLV_1;
notation TARSKI, XBOOLE_0, SUBSET_1, XCMPLX_0, XREAL_0, NAT_1, STRUCT_0,
BINOP_1, RELAT_1, FUNCT_1, FUNCT_2, RLVECT_1, GR_CY_1, VECTSP_1,
FINSEQ_1, GROUP_1, GROUP_2, GROUP_3, GROUP_6;
constructors DOMAIN_1, BINOP_1, GR_CY_1, GROUP_6, PARTFUN1, XCMPLX_0,
MEMBERED, XBOOLE_0;
clusters FUNCT_1, GROUP_1, GROUP_2, GROUP_3, GROUP_6, GR_CY_1, STRUCT_0,
RELSET_1, FINSEQ_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0;
requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM;

begin

reserve i,j,k for Nat;

definition let IT be Group;
attr IT is solvable means
:: GRSOLV_1:def 1

ex F being FinSequence of Subgroups IT st len F>0 & F.1=(Omega).IT &
F.(len F)=(1).IT
& for i st i in dom F & i+1 in dom F
for G1,G2 being strict Subgroup of IT st G1=F.i & G2=F.(i+1) holds
G2 is strict normal Subgroup of G1 &
for N being normal Subgroup of G1 st N=G2 holds G1./.N is commutative;
end;

definition
cluster solvable strict Group;
end;

theorem :: GRSOLV_1:1
for G being strict Group for H,F1,F2 being strict Subgroup of G st
F1 is normal Subgroup of F2 holds F1 /\ H is normal Subgroup of F2 /\ H;

theorem :: GRSOLV_1:2
for G being strict Group for F2 being strict Subgroup of G
for F1 being strict normal Subgroup of F2
for a,b being Element of F2
holds (a*F1) *(b*F1)=a *b *F1;

theorem :: GRSOLV_1:3
for G being strict Group for H,F2 being strict Subgroup of G
for F1 being strict normal Subgroup of F2 holds
for G2 being strict Subgroup of G st G2=H/\F2
for G1 being normal Subgroup of G2 st G1=H /\ F1
ex G3 being Subgroup of F2./.F1 st
G2./.G1, G3 are_isomorphic;

theorem :: GRSOLV_1:4
for G being strict Group for H,F2 being strict Subgroup of G
for F1 being strict normal Subgroup of F2 holds
for G2 being strict Subgroup of G st G2=F2/\H
for G1 being normal Subgroup of G2 st G1=F1/\H
ex G3 being Subgroup of F2./.F1 st
G2./.G1, G3 are_isomorphic;

theorem :: GRSOLV_1:5
for G being solvable strict Group for H being strict Subgroup of G holds
H is solvable;

theorem :: GRSOLV_1:6
for G being strict Group st
ex F being FinSequence of Subgroups G st
len F>0 & F.1=(Omega).G & F.(len F)=(1).G &
for i st i in dom F & i+1 in dom F
for G1,G2 being strict Subgroup of G st G1=F.i & G2=F.(i+1) holds
G2 is strict normal Subgroup of G1 &
for N being normal Subgroup of G1 st N=G2 holds G1./.N is cyclic Group
holds G is solvable;

theorem :: GRSOLV_1:7
for G being strict commutative Group holds G is strict solvable;

definition let G,H be strict Group;
let g be Homomorphism of G,H; let A be Subgroup of G;
func g|A -> Homomorphism of A,H equals
:: GRSOLV_1:def 2
g|(the carrier of A);
end;

definition let G,H be strict Group;
let g be Homomorphism of G,H;let A be Subgroup of G;
func g.:A -> strict Subgroup of H equals
:: GRSOLV_1:def 3
Image (g| A);
end;

theorem :: GRSOLV_1:8
for G,H being strict Group, g being Homomorphism of G,H
for A being Subgroup of G holds
rng (g|A)=g.:(the carrier of A);

theorem :: GRSOLV_1:9
for G,H being strict Group, g being Homomorphism of G,H
for A being strict Subgroup of G holds
the carrier of (g.: A)=g.:(the carrier of A);

theorem :: GRSOLV_1:10
for G,H being strict Group, h being Homomorphism of G,H holds
for A being strict Subgroup of G holds
Image(h|A) is strict Subgroup of Image h;

theorem :: GRSOLV_1:11
for G,H being strict Group, h being Homomorphism of G,H holds
for A being strict Subgroup of G holds h.:A is strict Subgroup of Image h;

theorem :: GRSOLV_1:12
for G,H being strict Group, h being Homomorphism of G,H holds
h.:((1).G)=(1).H & h.:((Omega).G)=(Omega).(Image h);

theorem :: GRSOLV_1:13
for G,H being strict Group, h being Homomorphism of G,H
for A,B being strict Subgroup of G holds
A is Subgroup of B implies
h.:A is Subgroup of h.:B;

theorem :: GRSOLV_1:14
for G,H being strict Group, h being Homomorphism of G,H
for A being strict Subgroup of G for a being Element of G
holds
h.a* h.:A=h.:(a*A) & h.:A * h.a=h.:(A*a);

theorem :: GRSOLV_1:15
for G,H being strict Group, h being Homomorphism of G,H
for A,B being Subset of G holds
h.:A* h.:B=h.:(A*B);

theorem :: GRSOLV_1:16
for G,H being strict Group, h being Homomorphism of G,H
for A,B being strict Subgroup of G holds
A is strict normal Subgroup of B implies
h.:A is strict normal Subgroup of h.:B;

theorem :: GRSOLV_1:17
for G,H being strict Group, h being Homomorphism of G,H holds
G is solvable Group implies Image h is solvable;
```