:: Solvable Groups :: by Katarzyna Zawadzka :: :: Received October 23, 1994 :: Copyright (c) 1994-2019 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 NUMBERS, SUBSET_1, GROUP_1, FINSEQ_1, GROUP_3, XXREAL_0, CARD_1, FUNCT_1, RLSUB_1, GROUP_2, RELAT_1, ARYTM_3, STRUCT_0, PRE_TOPC, GROUP_6, BINOP_1, TARSKI, XBOOLE_0, QC_LANG1, WELLORD1, NAT_1, FUNCT_2, ALGSTR_0, GRAPH_1, GRSOLV_1; notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, NAT_1, STRUCT_0, ALGSTR_0, BINOP_1, RELAT_1, FUNCT_1, RELSET_1, FUNCT_2, GR_CY_1, FINSEQ_1, GROUP_1, GROUP_2, GROUP_3, GROUP_6, XXREAL_0; constructors BINOP_1, FINSUB_1, REALSET1, GROUP_6, GR_CY_1, RELSET_1; registrations XBOOLE_0, FUNCT_1, RELSET_1, XREAL_0, FINSEQ_1, STRUCT_0, GROUP_2, GROUP_3, GROUP_6, GR_CY_1, ORDINAL1; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin reserve i for Element of 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; registration cluster solvable strict for Group; end; theorem :: GRSOLV_1:1 for G being Group, 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 Group for F2 being Subgroup of G for F1 being 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 Group for H,F2 being Subgroup of G for F1 being normal Subgroup of F2 holds for G2 being 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 Group for H,F2 being Subgroup of G for F1 being normal Subgroup of F2 holds for G2 being 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 strict solvable Group, H being strict Subgroup of G holds H is solvable; theorem :: GRSOLV_1:6 for G being 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 solvable; definition let G,H be 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 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 Group, g being Homomorphism of G,H for A being Subgroup of G holds the carrier of (g.: A) = g.:(the carrier of A); theorem :: GRSOLV_1:9 for G,H being 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_1:10 for G,H being 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_1:11 for G,H being Group, h being Homomorphism of G,H holds h.:((1).G)=(1).H & h.:((Omega).G)=(Omega).(Image h); theorem :: GRSOLV_1:12 for G,H being Group, h being Homomorphism of G,H for A,B being Subgroup of G holds A is Subgroup of B implies h.:A is Subgroup of h.:B; theorem :: GRSOLV_1:13 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_1:14 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_1:15 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_1:16 for G,H being strict Group, h being Homomorphism of G,H holds G is solvable Group implies Image h is solvable;