1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529
|
#############################################################################
##
#W grplatt.gi GAP library Martin Sch"onert,
#W Alexander Hulpke
##
#H @(#)$Id: grplatt.gi,v 4.70.2.1 2006/11/13 16:52:31 gap Exp $
##
#Y Copyright (C) 1996, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains declarations for subgroup latices
##
Revision.grplatt_gi:=
"@(#)$Id: grplatt.gi,v 4.70.2.1 2006/11/13 16:52:31 gap Exp $";
#############################################################################
##
#F Zuppos(<G>) . set of generators for cyclic subgroups of prime power size
##
InstallMethod(Zuppos,"group",true,[IsGroup],0,
function (G)
local zuppos, # set of zuppos,result
c, # a representative of a class of elements
o, # its order
N, # normalizer of < c >
t; # loop variable
# compute the zuppos
zuppos:=[One(G)];
for c in List(ConjugacyClasses(G),Representative) do
o:=Order(c);
if IsPrimePowerInt(o) then
if ForAll([2..o],i -> Gcd(o,i) <> 1 or not c^i in zuppos) then
N:=Normalizer(G,Subgroup(G,[c]));
for t in RightTransversal(G,N) do
Add(zuppos,c^t);
od;
fi;
fi;
od;
# return the set of zuppos
Sort(zuppos);
return zuppos;
end);
#############################################################################
##
#M ConjugacyClassSubgroups(<G>,<g>) . . . . . . . . . . . . constructor
##
InstallMethod(ConjugacyClassSubgroups,IsIdenticalObj,[IsGroup,IsGroup],0,
function(G,U)
local filter,cl;
if CanComputeSizeAnySubgroup(G) then
filter:=IsConjugacyClassSubgroupsByStabilizerRep;
else
filter:=IsConjugacyClassSubgroupsRep;
fi;
cl:=Objectify(NewType(CollectionsFamily(FamilyObj(G)),
filter),rec());
SetActingDomain(cl,G);
SetRepresentative(cl,U);
SetFunctionAction(cl,OnPoints);
return cl;
end);
#############################################################################
##
#M <clasa> = <clasb> . . . . . . . . . . . . . . . . . . by conjugacy test
##
InstallMethod( \=, IsIdenticalObj, [ IsConjugacyClassSubgroupsRep,
IsConjugacyClassSubgroupsRep ], 0,
function( clasa, clasb )
if not IsIdenticalObj(ActingDomain(clasa),ActingDomain(clasb))
then TryNextMethod();
fi;
return RepresentativeAction(ActingDomain(clasa),Representative(clasa),
Representative(clasb))<>fail;
end);
#############################################################################
##
#M <G> in <clas> . . . . . . . . . . . . . . . . . . by conjugacy test
##
InstallMethod( \in, IsElmsColls, [ IsGroup,IsConjugacyClassSubgroupsRep], 0,
function( G, clas )
return RepresentativeAction(ActingDomain(clas),Representative(clas),G)
<>fail;
end);
#############################################################################
##
#M AsList(<cls>)
##
InstallOtherMethod(AsList, "for classes of subgroups",
true, [ IsConjugacyClassSubgroupsRep],0,
function(c)
local rep;
rep:=Representative(c);
if not IsBound(c!.normalizerTransversal) then
c!.normalizerTransversal:=
RightTransversal(ActingDomain(c),StabilizerOfExternalSet(c));
fi;
if HasParent(rep) and IsSubset(Parent(rep),ActingDomain(c)) then
return List(c!.normalizerTransversal,i->ConjugateSubgroup(rep,i));
else
return List(c!.normalizerTransversal,i->ConjugateGroup(rep,i));
fi;
end);
#############################################################################
##
#M ClassElementLattice
##
InstallMethod(ClassElementLattice, "for classes of subgroups",
true, [ IsConjugacyClassSubgroupsRep, IsPosInt],0,
function(c,nr)
local rep;
rep:=Representative(c);
if not IsBound(c!.normalizerTransversal) then
c!.normalizerTransversal:=
RightTransversal(ActingDomain(c),StabilizerOfExternalSet(c));
fi;
return ConjugateSubgroup(rep,c!.normalizerTransversal[nr]);
end);
InstallOtherMethod( \[\], "for classes of subgroups",
true, [ IsConjugacyClassSubgroupsRep, IsPosInt],0,ClassElementLattice );
InstallMethod( StabilizerOfExternalSet, true, [ IsConjugacyClassSubgroupsRep ],
# override potential pc method
10,
function(xset)
return Normalizer(ActingDomain(xset),Representative(xset));
end);
InstallOtherMethod( NormalizerOp, true, [ IsConjugacyClassSubgroupsRep ], 0,
StabilizerOfExternalSet );
#############################################################################
##
#M PrintObj(<cl>) . . . . . . . . . . . . . . . . . . . . print function
##
InstallMethod(PrintObj,true,[IsConjugacyClassSubgroupsRep],0,
function(cl)
Print("ConjugacyClassSubgroups(",ActingDomain(cl),",",
Representative(cl),")");
end);
#############################################################################
##
#M ConjugacyClassesSubgroups(<G>) . classes of subgroups of a group
##
InstallMethod(ConjugacyClassesSubgroups,"group",true,[IsGroup],0,
function(G)
return ConjugacyClassesSubgroups(LatticeSubgroups(G));
end);
InstallOtherMethod(ConjugacyClassesSubgroups,"lattice",true,
[IsLatticeSubgroupsRep],0,
function(L)
return L!.conjugacyClassesSubgroups;
end);
#############################################################################
##
#F LatticeByCyclicExtension(<G>[,<func>[,<noperf>]]) Lattice of subgroups
##
## computes the lattice of <G> using the cyclic extension algorithm. If the
## function <func> is given, the algorithm will discard all subgroups not
## fulfilling <func> (and will also not extend them), returning a partial
## lattice. This can be useful to compute only subgroups with certain
## properties. Note however that this will *not* necessarily yield all
## subgroups that fulfill <func>, but the subgroups whose subgroups used
## for the construction also fulfill <func> as well.
##
# the following functions are declared only later
SOLVABILITY_IMPLYING_FUNCTIONS:=
[IsSolvableGroup,IsNilpotentGroup,IsPGroup,IsCyclic];
InstallGlobalFunction( LatticeByCyclicExtension, function(arg)
local G, # group
func, # test function
noperf, # discard perfect groups
lattice, # lattice (result)
factors, # factorization of <G>'s size
zuppos, # generators of prime power order
zupposPrime, # corresponding prime
zupposPower, # index of power of generator
ZupposSubgroup, # function to compute zuppos for subgroup
zuperms, # permutation of zuppos by group
Gimg, # grp image under zuperms
nrClasses, # number of classes
classes, # list of all classes
classesZups, # zuppos blist of classes
classesExts, # extend-by blist of classes
perfect, # classes of perfect subgroups of <G>
perfectNew, # this class of perfect subgroups is new
perfectZups, # zuppos blist of perfect subgroups
layerb, # begin of previous layer
layere, # end of previous layer
H, # representative of a class
Hzups, # zuppos blist of <H>
Hexts, # extend blist of <H>
C, # class of <I>
I, # new subgroup found
Ielms, # elements of <I>
Izups, # zuppos blist of <I>
N, # normalizer of <I>
Nzups, # zuppos blist of <N>
Jzups, # zuppos of a conjugate of <I>
Kzups, # zuppos of a representative in <classes>
reps, # transversal of <N> in <G>
ac,
transv,
factored,
mapped,
expandmem,
h,i,k,l,ri,rl,r; # loop variables
G:=arg[1];
noperf:=false;
if Length(arg)>1 and IsFunction(arg[2]) then
func:=arg[2];
Info(InfoLattice,1,"lattice discarding function active!");
if Length(arg)>2 and IsBool(arg[3]) then
noperf:=arg[3];
fi;
else
func:=false;
fi;
expandmem:=ValueOption("Expand")=true;
# if store is true, an element list will be kept in `Ielms' if possible
ZupposSubgroup:=function(U,store)
local elms,zups;
if Size(U)=Size(G) then
if store then Ielms:=fail;fi;
zups:=BlistList([1..Length(zuppos)],[1..Length(zuppos)]);
elif Size(U)>10^4 then
# the group is very big - test the zuppos with `in'
Info(InfoLattice,3,"testing zuppos with `in'");
if store then Ielms:=fail;fi;
zups:=List(zuppos,i->i in U);
IsBlist(zups);
else
elms:=AsSSortedListNonstored(U);
if store then Ielms:=elms;fi;
zups:=BlistList(zuppos,elms);
fi;
return zups;
end;
# compute the factorized size of <G>
factors:=Factors(Size(G));
# compute a system of generators for the cyclic sgr. of prime power size
zuppos:=Zuppos(G);
Info(InfoLattice,1,"<G> has ",Length(zuppos)," zuppos");
# compute zuppo permutation
if IsPermGroup(G) then
zuppos:=List(zuppos,SmallestGeneratorPerm);
zuppos:=AsSSortedList(zuppos);
zuperms:=List(GeneratorsOfGroup(G),
i->Permutation(i,zuppos,function(x,a)
return SmallestGeneratorPerm(x^a);
end));
if NrMovedPoints(zuperms)<200*NrMovedPoints(G) then
zuperms:=GroupHomomorphismByImagesNC(G,Group(zuperms),
GeneratorsOfGroup(G),zuperms);
# force kernel, also enforces injective setting
Gimg:=Image(zuperms);
if Size(KernelOfMultiplicativeGeneralMapping(zuperms))=1 then
SetSize(Gimg,Size(G));
fi;
else
zuperms:=fail;
fi;
else
zuppos:=AsSSortedList(zuppos);
zuperms:=fail;
fi;
# compute the prime corresponding to each zuppo and the index of power
zupposPrime:=[];
zupposPower:=[];
for r in zuppos do
i:=SmallestRootInt(Order(r));
Add(zupposPrime,i);
k:=0;
while k <> false do
k:=k + 1;
if GcdInt(i,k) = 1 then
l:=Position(zuppos,r^(i*k));
if l <> fail then
Add(zupposPower,l);
k:=false;
fi;
fi;
od;
od;
Info(InfoLattice,1,"powers computed");
if func<>false and
(noperf or func in SOLVABILITY_IMPLYING_FUNCTIONS) then
Info(InfoLattice,1,"Ignoring perfect subgroups");
perfect:=[];
else
if IsPermGroup(G) then
# trigger potentially better methods
IsNaturalSymmetricGroup(G);
IsNaturalAlternatingGroup(G);
fi;
perfect:=RepresentativesPerfectSubgroups(G);
perfect:=Filtered(perfect,i->Size(i)>1 and Size(i)<Size(G));
if func<>false then
perfect:=Filtered(perfect,func);
fi;
perfect:=List(perfect,i->AsSubgroup(Parent(G),i));
fi;
perfectZups:=[];
perfectNew :=[];
for i in [1..Length(perfect)] do
I:=perfect[i];
#perfectZups[i]:=BlistList(zuppos,AsSSortedListNonstored(I));
perfectZups[i]:=ZupposSubgroup(I,false);
perfectNew[i]:=true;
od;
Info(InfoLattice,1,"<G> has ",Length(perfect),
" representatives of perfect subgroups");
# initialize the classes list
nrClasses:=1;
classes:=ConjugacyClassSubgroups(G,TrivialSubgroup(G));
SetSize(classes,1);
classes:=[classes];
classesZups:=[BlistList(zuppos,[One(G)])];
classesExts:=[DifferenceBlist(BlistList(zuppos,zuppos),classesZups[1])];
layerb:=1;
layere:=1;
# loop over the layers of group (except the group itself)
for l in [1..Length(factors)-1] do
Info(InfoLattice,1,"doing layer ",l,",",
"previous layer has ",layere-layerb+1," classes");
# extend representatives of the classes of the previous layer
for h in [layerb..layere] do
# get the representative,its zuppos blist and extend-by blist
H:=Representative(classes[h]);
Hzups:=classesZups[h];
Hexts:=classesExts[h];
Info(InfoLattice,2,"extending subgroup ",h,", size = ",Size(H));
# loop over the zuppos whose <p>-th power lies in <H>
for i in [1..Length(zuppos)] do
if Hexts[i] and Hzups[zupposPower[i]] then
# make the new subgroup <I>
# NC is safe -- all groups are subgroups of Parent(H)
I:=ClosureSubgroupNC(H,zuppos[i]);
#Subgroup(Parent(G),Concatenation(GeneratorsOfGroup(H),
# [zuppos[i]]));
if func=false or func(I) then
SetSize(I,Size(H) * zupposPrime[i]);
# compute the zuppos blist of <I>
#Ielms:=AsSSortedListNonstored(I);
#Izups:=BlistList(zuppos,Ielms);
if zuperms=fail then
Izups:=ZupposSubgroup(I,true);
else
Izups:=ZupposSubgroup(I,false);
fi;
# compute the normalizer of <I>
N:=Normalizer(G,I);
#AH 'NormalizerInParent' attribute ?
Info(InfoLattice,2,"found new class ",nrClasses+1,
", size = ",Size(I)," length = ",Size(G)/Size(N));
# make the new conjugacy class
C:=ConjugacyClassSubgroups(G,I);
SetSize(C,Size(G) / Size(N));
SetStabilizerOfExternalSet(C,N);
nrClasses:=nrClasses + 1;
classes[nrClasses]:=C;
# store the extend by list
if l < Length(factors)-1 then
classesZups[nrClasses]:=Izups;
#Nzups:=BlistList(zuppos,AsSSortedListNonstored(N));
Nzups:=ZupposSubgroup(N,false);
SubtractBlist(Nzups,Izups);
classesExts[nrClasses]:=Nzups;
fi;
# compute the right transversal
# (but don't store it in the parent)
if expandmem and zuperms<>fail then
if Index(G,N)>400 then
ac:=AscendingChainOp(G,N); # do not store
while Length(ac)>2 and Index(ac[3],ac[1])<100 do
ac:=Concatenation([ac[1]],ac{[3..Length(ac)]});
od;
if Length(ac)>2 and
Maximum(List([3..Length(ac)],x->Index(ac[x],ac[x-1])))<500
then
# mapped factorized transversal
Info(InfoLattice,3,"factorized transversal ",
List([2..Length(ac)],x->Index(ac[x],ac[x-1])));
transv:=[];
ac[Length(ac)]:=Gimg;
for ri in [Length(ac)-1,Length(ac)-2..1] do
ac[ri]:=Image(zuperms,ac[ri]);
if ri=1 then
transv[ri]:=List(RightTransversalOp(ac[ri+1],ac[ri]),
i->Permuted(Izups,i));
else
transv[ri]:=AsList(RightTransversalOp(ac[ri+1],ac[ri]));
fi;
od;
mapped:=true;
factored:=true;
reps:=Cartesian(transv);
Unbind(ac);
Unbind(transv);
else
reps:=RightTransversalOp(Gimg,Image(zuperms,N));
mapped:=true;
factored:=false;
fi;
else
reps:=RightTransversalOp(G,N);
mapped:=false;
factored:=false;
fi;
else
reps:=RightTransversalOp(G,N);
mapped:=false;
factored:=false;
fi;
# loop over the conjugates of <I>
for ri in [1..Length(reps)] do
CompletionBar(InfoLattice,3,"Coset loop: ",ri/Length(reps));
r:=reps[ri];
# compute the zuppos blist of the conjugate
if zuperms<>fail then
# we know the permutation of zuppos by the group
if mapped then
if factored then
Jzups:=r[1];
for rl in [2..Length(r)] do
Jzups:=Permuted(Jzups,r[rl]);
od;
else
Jzups:=Permuted(Izups,r);
fi;
else
if factored then
Error("factored");
else
Jzups:=Image(zuperms,r);
Jzups:=Permuted(Izups,Jzups);
fi;
fi;
elif r = One(G) then
Jzups:=Izups;
elif Ielms<>fail then
Jzups:=BlistList(zuppos,OnTuples(Ielms,r));
else
Jzups:=ZupposSubgroup(I^r,false);
fi;
# loop over the already found classes
for k in [h..layere] do
Kzups:=classesZups[k];
# test if the <K> is a subgroup of <J>
if IsSubsetBlist(Jzups,Kzups) then
# don't extend <K> by the elements of <J>
SubtractBlist(classesExts[k],Jzups);
fi;
od;
od;
CompletionBar(InfoLattice,3,"Coset loop: ",false);
# now we are done with the new class
Unbind(Ielms);
Unbind(reps);
Info(InfoLattice,2,"tested inclusions");
else
Info(InfoLattice,1,"discarded!");
fi; # if condition fulfilled
fi; # if Hexts[i] and Hzups[zupposPower[i]] then ...
od; # for i in [1..Length(zuppos)] do ...
# remove the stuff we don't need any more
Unbind(classesZups[h]);
Unbind(classesExts[h]);
od; # for h in [layerb..layere] do ...
# add the classes of perfect subgroups
for i in [1..Length(perfect)] do
if perfectNew[i]
and IsPerfectGroup(perfect[i])
and Length(Factors(Size(perfect[i]))) = l
then
# make the new subgroup <I>
I:=perfect[i];
# compute the zuppos blist of <I>
#Ielms:=AsSSortedListNonstored(I);
#Izups:=BlistList(zuppos,Ielms);
if zuperms=fail then
Izups:=ZupposSubgroup(I,true);
else
Izups:=ZupposSubgroup(I,false);
fi;
# compute the normalizer of <I>
N:=Normalizer(G,I);
# AH: NormalizerInParent ?
Info(InfoLattice,2,"found perfect class ",nrClasses+1,
" size = ",Size(I),", length = ",Size(G)/Size(N));
# make the new conjugacy class
C:=ConjugacyClassSubgroups(G,I);
SetSize(C,Size(G)/Size(N));
SetStabilizerOfExternalSet(C,N);
nrClasses:=nrClasses + 1;
classes[nrClasses]:=C;
# store the extend by list
if l < Length(factors)-1 then
classesZups[nrClasses]:=Izups;
#Nzups:=BlistList(zuppos,AsSSortedListNonstored(N));
Nzups:=ZupposSubgroup(N,false);
SubtractBlist(Nzups,Izups);
classesExts[nrClasses]:=Nzups;
fi;
# compute the right transversal
# (but don't store it in the parent)
reps:=RightTransversalOp(G,N);
# loop over the conjugates of <I>
for r in reps do
# compute the zuppos blist of the conjugate
if zuperms<>fail then
# we know the permutation of zuppos by the group
Jzups:=Image(zuperms,r);
Jzups:=Permuted(Izups,Jzups);
elif r = One(G) then
Jzups:=Izups;
elif Ielms<>fail then
Jzups:=BlistList(zuppos,OnTuples(Ielms,r));
else
Jzups:=ZupposSubgroup(I^r,false);
fi;
# loop over the perfect classes
for k in [i+1..Length(perfect)] do
Kzups:=perfectZups[k];
# throw away classes that appear twice in perfect
if Jzups = Kzups then
perfectNew[k]:=false;
perfectZups[k]:=[];
fi;
od;
od;
# now we are done with the new class
Unbind(Ielms);
Unbind(reps);
Info(InfoLattice,2,"tested equalities");
# unbind the stuff we dont need any more
perfectZups[i]:=[];
fi;
# if IsPerfectGroup(I) and Length(Factors(Size(I))) = layer the...
od; # for i in [1..Length(perfect)] do
# on to the next layer
layerb:=layere+1;
layere:=nrClasses;
od; # for l in [1..Length(factors)-1] do ...
# add the whole group to the list of classes
Info(InfoLattice,1,"doing layer ",Length(factors),",",
" previous layer has ",layere-layerb+1," classes");
if Size(G)>1 and (func=false or func(G)) then
Info(InfoLattice,2,"found whole group, size = ",Size(G),",","length = 1");
C:=ConjugacyClassSubgroups(G,G);
SetSize(C,1);
nrClasses:=nrClasses + 1;
classes[nrClasses]:=C;
fi;
# return the list of classes
Info(InfoLattice,1,"<G> has ",nrClasses," classes,",
" and ",Sum(classes,Size)," subgroups");
# sort the classes
Sort(classes,
function (c,d)
return Size(Representative(c)) < Size(Representative(d))
or (Size(Representative(c)) = Size(Representative(d))
and Size(c) < Size(d));
end);
# create the lattice
lattice:=Objectify(NewType(FamilyObj(classes),IsLatticeSubgroupsRep),
rec());
lattice!.conjugacyClassesSubgroups:=classes;
lattice!.group :=G;
if func<>false then
lattice!.func:=func;
fi;
# return the lattice
return lattice;
end );
#############################################################################
##
#M LatticeSubgroups(<G>) . . . . . . . . . . lattice of subgroups
##
InstallMethod(LatticeSubgroups,"cyclic extension",true,[IsGroup],0,
LatticeByCyclicExtension);
#############################################################################
##
#M Print for lattice
##
InstallMethod(ViewObj,"lattice",true,[IsLatticeSubgroupsRep],0,
function(l)
Print("<subgroup lattice of ");
ViewObj(l!.group);
Print(", ", Length(l!.conjugacyClassesSubgroups)," classes, ",
Sum(l!.conjugacyClassesSubgroups,Size)," subgroups");
if IsBound(l!.func) then
Print(", restricted under further condition l!.func");
fi;
Print(">");
end);
InstallMethod(PrintObj,"lattice",true,[IsLatticeSubgroupsRep],0,
function(l)
Print("LatticeSubgroups(",l!.group);
if IsBound(l!.func) then
Print("),# under further condition l!.func\n");
else
Print(")");
fi;
end);
#############################################################################
##
#M ConjugacyClassesPerfectSubgroups
##
InstallMethod(ConjugacyClassesPerfectSubgroups,"generic",true,[IsGroup],0,
function(G)
return
List(RepresentativesPerfectSubgroups(G),i->ConjugacyClassSubgroups(G,i));
end);
#############################################################################
##
#M PerfectResiduum
##
InstallMethod(PerfectResiduum,"for groups",true,
[IsGroup],0,
function(G)
while not IsPerfectGroup(G) do
G:=DerivedSubgroup(G);
od;
return G;
end);
#############################################################################
##
#M RepresentativesPerfectSubgroups solvable
##
InstallMethod(RepresentativesPerfectSubgroups,"solvable",true,
[IsSolvableGroup],0,
function(G)
return [TrivialSubgroup(G)];
end);
#############################################################################
##
#M RepresentativesPerfectSubgroups
##
BindGlobal("RepsPerfSimpSub",function(G,simple)
local badsizes,n,un,cl,r,i,l,u,bw,cnt,gens,go,imgs,bg,bi,emb,nu,k,j,
D,params,might,bo;
if IsSolvableGroup(G) then
return [TrivialSubgroup(G)];
else
PerfGrpLoad(0);
badsizes := Union(PERFRec.notAvailable,PERFRec.notKnown);
D:=G;
D:=PerfectResiduum(D);
n:=Size(D);
Info(InfoLattice,1,"The perfect residuum has size ",n);
# sizes of possible perfect subgroups
un:=Filtered(DivisorsInt(n),i->i>1
# index <=4 would lead to solvable factor
and i<n/4);
# if D is simple, we can limit indices further
if IsSimpleGroup(D) then
k:=4;
l:=120;
while l<n do
k:=k+1;
l:=l*(k+1);
od;
# now k is maximal such that k!<Size(D). Thus subgroups of D must have
# index more than k
k:=Int(n/k);
un:=Filtered(un,i->i<=k);
fi;
Info(InfoLattice,1,"Searching perfect groups up to size ",Maximum(un));
if ForAny(un,i->i>10^6) then
Error("the perfect residuum is too large");
fi;
un:=Filtered(un,i->i in PERFRec.sizes);
if Length(Intersection(badsizes,un))>0 then
Error(
"failed due to incomplete information in the Holt/Plesken library");
fi;
cl:=Filtered(ConjugacyClasses(G),i->Representative(i) in D);
Info(InfoLattice,2,Length(cl)," classes of ",
Length(ConjugacyClasses(G))," to consider");
r:=[];
for i in un do
l:=NumberPerfectGroups(i);
if l>0 then
for j in [1..l] do
u:=PerfectGroup(IsPermGroup,i,j);
Info(InfoLattice,1,"trying group ",i,",",j,": ",u);
# test whether there is a chance to embed
might:=simple=false or IsSimpleGroup(u);
cnt:=0;
while might and cnt<20 do
bg:=Order(Random(u));
might:=ForAny(cl,i->Order(Representative(i))=bg);
cnt:=cnt+1;
od;
if might then
# find a suitable generating system
bw:=infinity;
bo:=[0,0];
cnt:=0;
repeat
if cnt=0 then
# first the small gen syst.
gens:=SmallGeneratingSet(u);
else
# then something random
repeat
if Length(gens)>2 and Random([1,2])=1 then
# try to get down to 2 gens
gens:=List([1,2],i->Random(u));
else
gens:=List([1..Random([2..Length(SmallGeneratingSet(u))])],
i->Random(u));
fi;
# try to get small orders
for k in [1..Length(gens)] do
go:=Order(gens[k]);
# try a p-element
if Random([1..2*Length(gens)])=1 then
gens[k]:=gens[k]^(go/(Random(Factors(go))));
fi;
od;
until Index(u,SubgroupNC(u,gens))=1;
fi;
go:=List(gens,Order);
imgs:=List(go,i->Filtered(cl,j->Order(Representative(j))=i));
Info(InfoLattice,3,go,":",Product(imgs,i->Sum(i,Size)));
if Product(imgs,i->Sum(i,Size))<bw then
bg:=gens;
bo:=go;
bi:=imgs;
bw:=Product(imgs,i->Sum(i,Size));
elif Set(go)=Set(bo) then
# we hit the orders again -> sign that we can't be
# completely off track
cnt:=cnt+Int(bw/Size(G)*3);
fi;
cnt:=cnt+1;
until bw/Size(G)*6<cnt;
if bw>0 then
Info(InfoLattice,2,"find ",bw," from ",cnt);
# find all embeddings
params:=rec(gens:=bg,from:=u);
emb:=MorClassLoop(G,bi,params,
# all injective homs = 1+2+8
11);
#emb:=MorClassLoop(G,bi,rec(type:=2,what:=3,gens:=bg,from:=u,
# elms:=false,size:=Size(u)));
Info(InfoLattice,2,Length(emb)," embeddings");
nu:=[];
for k in emb do
k:=Image(k,u);
if not ForAny(nu,i->RepresentativeAction(G,i,k)<>fail) then
Add(nu,k);
k!.perfectType:=[i,j];
fi;
od;
Info(InfoLattice,1,Length(nu)," classes");
r:=Concatenation(r,nu);
fi;
else
Info(InfoLattice,2,"cannot embed");
fi;
od;
fi;
od;
# add the two obvious ones
Add(r,D);
Add(r,TrivialSubgroup(G));
return r;
fi;
end);
InstallMethod(RepresentativesPerfectSubgroups,"using Holt/Plesken library",
true,[IsGroup],0,G->RepsPerfSimpSub(G,false));
InstallMethod(RepresentativesSimpleSubgroups,"using Holt/Plesken library",
true,[IsGroup],0,G->RepsPerfSimpSub(G,true));
InstallMethod(RepresentativesSimpleSubgroups,"if perfect subs are known",
true,[IsGroup and HasRepresentativesPerfectSubgroups],0,
G->Filtered(RepresentativesPerfectSubgroups(G),IsSimpleGroup));
#############################################################################
##
#M MaximalSubgroupsLattice
##
InstallMethod(MaximalSubgroupsLattice,"cyclic extension",true,
[IsLatticeSubgroupsRep],0,
function (L)
local maximals, # maximals as pair <class>,<conj> (result)
maximalsZups, # their zuppos blist
cnt, # count for information messages
zuppos, # generators of prime power order
classes, # list of all classes
classesZups, # zuppos blist of classes
I, # representative of a class
Ielms, # elements of <I>
Izups, # zuppos blist of <I>
N, # normalizer of <I>
Jzups, # zuppos of a conjugate of <I>
Kzups, # zuppos of a representative in <classes>
reps, # transversal of <N> in <G>
grp, # the group
i,k,r; # loop variables
if IsBound(L!.func) then
Error("cannot compute maximality inclusions for partial lattice");
fi;
grp:=L!.group;
# compute the lattice,fetch the classes,zuppos,and representatives
classes:=L!.conjugacyClassesSubgroups;
classesZups:=[];
# compute a system of generators for the cyclic sgr. of prime power size
zuppos:=Zuppos(grp);
# initialize the maximals list
Info(InfoLattice,1,"computing maximal relationship");
maximals:=List(classes,c -> []);
maximalsZups:=List(classes,c -> []);
# find the minimal supergroups of the whole group
Info(InfoLattice,2,"testing class ",Length(classes),", size = ",
Size(grp),", length = 1, included in 0 minimal subs");
classesZups[Length(classes)]:=BlistList(zuppos,zuppos);
# loop over all classes
for i in [Length(classes)-1,Length(classes)-2..1] do
# take the subgroup <I>
I:=Representative(classes[i]);
Info(InfoLattice,2," testing class ",i);
# compute the zuppos blist of <I>
Ielms:=AsSSortedListNonstored(I);
Izups:=BlistList(zuppos,Ielms);
classesZups[i]:=Izups;
# compute the normalizer of <I>
N:=Normalizer(grp,I);
# compute the right transversal
# (but don't store it in the parent)
reps:=RightTransversalOp(grp,N);
# initialize the counter
cnt:=0;
# loop over the conjugates of <I>
for r in [1..Length(reps)] do
# compute the zuppos blist of the conjugate
if reps[r] = One(grp) then
Jzups:=Izups;
else
Jzups:=BlistList(zuppos,OnTuples(Ielms,reps[r]));
fi;
# loop over all other (larger classes)
for k in [i+1..Length(classes)] do
Kzups:=classesZups[k];
# test if the <K> is a minimal supergroup of <J>
if IsSubsetBlist(Kzups,Jzups)
and ForAll(maximalsZups[k],
zups -> not IsSubsetBlist(zups,Jzups))
then
Add(maximals[k],[ i,r ]);
Add(maximalsZups[k],Jzups);
cnt:=cnt + 1;
fi;
od;
od;
# inform about the count
Unbind(Ielms);
Unbind(reps);
Info(InfoLattice,2,"size = ",Size(I),", length = ",
Size(grp) / Size(N),", included in ",cnt," minimal sups");
od;
return maximals;
end);
#############################################################################
##
#M MinimalSupergroupsLattice
##
InstallMethod(MinimalSupergroupsLattice,"cyclic extension",true,
[IsLatticeSubgroupsRep],0,
function (L)
local minimals, # minimals as pair <class>,<conj> (result)
minimalsZups, # their zuppos blist
cnt, # count for information messages
zuppos, # generators of prime power order
classes, # list of all classes
classesZups, # zuppos blist of classes
I, # representative of a class
Ielms, # elements of <I>
Izups, # zuppos blist of <I>
N, # normalizer of <I>
Jzups, # zuppos of a conjugate of <I>
Kzups, # zuppos of a representative in <classes>
reps, # transversal of <N> in <G>
grp, # the group
i,k,r; # loop variables
if IsBound(L!.func) then
Error("cannot compute maximality inclusions for partial lattice");
fi;
grp:=L!.group;
# compute the lattice,fetch the classes,zuppos,and representatives
classes:=L!.conjugacyClassesSubgroups;
classesZups:=[];
# compute a system of generators for the cyclic sgr. of prime power size
zuppos:=Zuppos(grp);
# initialize the minimals list
Info(InfoLattice,1,"computing minimal relationship");
minimals:=List(classes,c -> []);
minimalsZups:=List(classes,c -> []);
# loop over all classes
for i in [1..Length(classes)-1] do
# take the subgroup <I>
I:=Representative(classes[i]);
# compute the zuppos blist of <I>
Ielms:=AsSSortedListNonstored(I);
Izups:=BlistList(zuppos,Ielms);
classesZups[i]:=Izups;
# compute the normalizer of <I>
N:=Normalizer(grp,I);
# compute the right transversal (but don't store it in the parent)
reps:=RightTransversalOp(grp,N);
# initialize the counter
cnt:=0;
# loop over the conjugates of <I>
for r in [1..Length(reps)] do
# compute the zuppos blist of the conjugate
if reps[r] = One(grp) then
Jzups:=Izups;
else
Jzups:=BlistList(zuppos,OnTuples(Ielms,reps[r]));
fi;
# loop over all other (smaller classes)
for k in [1..i-1] do
Kzups:=classesZups[k];
# test if the <K> is a maximal subgroup of <J>
if IsSubsetBlist(Jzups,Kzups)
and ForAll(minimalsZups[k],
zups -> not IsSubsetBlist(Jzups,zups))
then
Add(minimals[k],[ i,r ]);
Add(minimalsZups[k],Jzups);
cnt:=cnt + 1;
fi;
od;
od;
# inform about the count
Unbind(Ielms);
Unbind(reps);
Info(InfoLattice,2,"testing class ",i,", size = ",Size(I),
", length = ",Size(grp) / Size(N),", includes ",cnt,
" maximal subs");
od;
# find the maximal subgroups of the whole group
cnt:=0;
for k in [1..Length(classes)-1] do
if minimals[k] = [] then
Add(minimals[k],[ Length(classes),1 ]);
cnt:=cnt + 1;
fi;
od;
Info(InfoLattice,2,"testing class ",Length(classes),", size = ",
Size(grp),", length = 1, includes ",cnt," maximal subs");
return minimals;
end);
#############################################################################
##
#F MaximalSubgroupClassReps(<G>) . . . . reps of conjugacy classes of
#F maximal subgroups
##
InstallMethod(MaximalSubgroupClassReps,"using lattice",true,[IsGroup],0,
function (G)
local maxs,lat;
#AH special AG treatment
if not HasIsSolvableGroup(G) and IsSolvableGroup(G) then
return MaximalSubgroupClassReps(G);
fi;
# simply compute all conjugacy classes and take the maximals
lat:=LatticeSubgroups(G);
maxs:=MaximalSubgroupsLattice(lat)[Length(lat!.conjugacyClassesSubgroups)];
maxs:=List(lat!.conjugacyClassesSubgroups{
Set(maxs{[1..Length(maxs)]}[1])},Representative);
return maxs;
end);
#############################################################################
##
#F ConjugacyClassesMaximalSubgroups(<G>)
##
InstallMethod(ConjugacyClassesMaximalSubgroups,
"use MaximalSubgroupClassReps",true,[IsGroup],0,
function(G)
return List(MaximalSubgroupClassReps(G),i->ConjugacyClassSubgroups(G,i));
end);
#############################################################################
##
#F MaximalSubgroups(<G>)
##
InstallMethod(MaximalSubgroups,
"expand list",true,[IsGroup],0,
function(G)
return Concatenation(List(ConjugacyClassesMaximalSubgroups(G),AsList));
end);
#############################################################################
##
#F NormalSubgroupsCalc(<G>[,<onlysimple>]) normal subs for pc or perm groups
##
NormalSubgroupsCalc := function (arg)
local G, # group
onlysimple, # determine only subgroups with simple composition factors
nt,nnt, # normal subgroups
cs, # comp. series
M,N, # nt . in series
mpcgs, # modulo pcgs
p, # prime
ocr, # 1-cohomology record
l, # list
vs, # vector space
hom, # homomorphism
jg, # generator images
auts, # factor automorphisms
T,S,C,A,ji,orb,orbi,cllen,r,o,c,inv,cnt,
i,j,k; # loop
G:=arg[1];
onlysimple:=false;
if Length(arg)>1 and arg[2]=true then
onlysimple:=true;
fi;
nt:=[G];
cs:=ChiefSeries(G);
for i in [2..Length(cs)] do
# we assume that nt contains all normal subgroups above cs[i-1]
# we want to lift to G/cs[i]
M:=cs[i-1];
N:=cs[i];
# the normal subgroups already known
if (not onlysimple) or (not HasAbelianFactorGroup(M,N)) then
nnt:=ShallowCopy(nt);
else
nnt:=[];
fi;
Info(InfoLattice,1,i,":",Index(M,N));
if HasAbelianFactorGroup(M,N) then
# the modulo pcgs
mpcgs:=ModuloPcgs(M,N);
p:=RelativeOrderOfPcElement(mpcgs,mpcgs[1]);
for j in Filtered(nt,i->Size(i)>Size(M)) do
# test centrality
if ForAll(GeneratorsOfGroup(j),
i->ForAll(mpcgs,j->Comm(i,j) in N)) then
Info(InfoLattice,2,"factorsize=",Index(j,N),"/",Index(M,N));
# reasons not to go complements
if (HasAbelianFactorGroup(j,N) and
p^(Length(mpcgs)*LogInt(Index(j,M),p))>100)
then
l:=fail; # we will compute the subgroups later
else
ocr:=rec(
group:=j,
modulePcgs:=mpcgs
);
# we want only normal complements. Therefore the 1-Coboundaries must
# be trivial. We compute these first.
if Dimension(OCOneCoboundaries(ocr))=0 then
l:=[];
OCOneCocycles(ocr,true);
if IsBound(ocr.complement) then
l:=BaseSteinitzVectors(BasisVectors(Basis(ocr.oneCocycles)),
BasisVectors(Basis(ocr.oneCoboundaries)));
vs:=VectorSpace(LeftActingDomain(ocr.oneCocycles),
l.factorspace,Zero(ocr.oneCocycles));
Info(InfoLattice,2,p^Length(l.factorspace)," cocycles");
# try to catch some solvable cases that look awful
if Size(vs)>1000 and Length(Set(Factors(Index(j,N))))<=2
then
l:=fail;
else
l:=[];
for k in vs do
k:=ClosureGroup(N,ocr.cocycleToComplement(k));
if IsNormal(G,k) then
Add(l,k);
fi;
od;
Info(InfoLattice,2," -> ",Length(l)," normal complements");
nnt:=Concatenation(nnt,l);
fi;
fi;
fi;
fi;
if l=fail then
if onlysimple then
# all groups obtained will have a solvable factor
l:=[];
else
Info(InfoLattice,1,"using invariant subgroups");
# the factor is abelian, we therefore find this homomorphism
# quick.
hom:=NaturalHomomorphismByNormalSubgroup(j,N);
r:=Image(hom,j);
jg:=List(GeneratorsOfGroup(j),i->Image(hom,i));
# construct the automorphisms
auts:=List(GeneratorsOfGroup(G),
i->GroupHomomorphismByImagesNC(r,r,jg,
List(GeneratorsOfGroup(j),k->Image(hom,k^i))));
l:=SubgroupsSolvableGroup(r,rec(
actions:=auts,
funcnorm:=r,
consider:=ExactSizeConsiderFunction(Index(j,M)),
normal:=true));
Info(InfoLattice,2,"found ",Length(l)," invariant subgroups");
C:=Image(hom,M);
l:=Filtered(l,i->Size(i)=Index(j,M) and Size(Intersection(i,C))=1);
l:=List(l,i->PreImage(hom,i));
l:=Filtered(l,i->IsNormal(G,i));
Info(InfoLattice,1,Length(l)," of these normal");
nnt:=Concatenation(nnt,l);
fi;
fi;
fi;
od;
else
# nonabelian factor.
# 1) compute the action for the factor
# first, we obtain the simple factors T_i/N.
# we get these as intersections of the conjugates of the subnormal
# subgroup
if HasCompositionSeries(M) then
T:=CompositionSeries(M)[2]; # stored attribute
else
T:=false;
fi;
if not (T<>false and IsSubgroup(T,N)) then
# we did not get the right T: must compute
hom:=NaturalHomomorphismByNormalSubgroup(M,N);
T:=CompositionSeries(Image(hom))[2];
T:=PreImage(hom,T);
fi;
hom:=NaturalHomomorphismByNormalSubgroup(M,T);
A:=Image(hom,M);
Info(InfoLattice,2,"Search involution");
# find involution in M/T
repeat
repeat
inv:=Random(M);
until (Order(inv) mod 2 =0) and not inv in T;
o:=First([2..Order(inv)],i->inv^i in T);
until (o mod 2 =0);
Info(InfoLattice,2,"Element of order ",o);
inv:=inv^(o/2); # this is an involution in the factor
Assert(1,inv^2 in T and not inv in T);
S:=Normalizer(G,T); # stabilize first component
orb:=[inv]; # class representatives in A by preimages in G
orbi:=[Image(hom,inv)];
cllen:=Index(A,Centralizer(A,orbi[1]));
C:=T; #starting centralizer
cnt:=1;
# we have to find at least 1 centralizing element
repeat
# find element that centralizes inv modulo T
repeat
r:=Random(S);
c:=Comm(inv,r);
o:=First([1..Order(c)],i->c^i in T);
c:=c^QuoInt(o-1,2);
if o mod 2=1 then
c:=r*c;
else
c:=inv^r*c;
fi;
# take care of potential class fusion
if not c in T and c in C then
cnt:=cnt+1;
if cnt=10 then
# if we have 10 true centralizing elements that did not
# yield anything new, we assume that classes get fused.
# So we have to test, how much fusion takes place.
# We do this with an orbit algorithm on classes of A
for j in orb do
for k in SmallGeneratingSet(S) do
j:=j^k;
ji:=Image(hom,j);
if ForAll(orbi,l->RepresentativeAction(A,l,ji)=fail) then
Add(orb,j);
Add(orbi,ji);
fi;
od;
od;
# now we have the length
cllen:=cllen*Length(orb);
Info(InfoLattice,1,Length(orb)," classes fuse");
fi;
fi;
until not c in C or Index(S,C)=cllen;
C:=ClosureGroup(C,c);
Info(InfoLattice,2,"New centralizing element of order ",o,
", Index=",Index(S,C));
until Index(S,C)<=cllen;
C:=Core(G,C); #the true centralizer is the core of the involution
# centralizer
if Size(C)>Size(N) then
for j in Filtered(nt,i->Size(i)>Size(M)) do
j:=Intersection(C,j);
if Size(j)>Size(N) and not j in nnt then
Add(nnt,j);
fi;
od;
fi;
fi; # else nonabelian
# the kernel itself
Add(nnt,N);
if onlysimple then
c:=Length(nnt);
nnt:=Filtered(nnt,j->Size(ClosureGroup(N,DerivedSubgroup(j)))=Size(j) );
Info(InfoLattice,2,"removed ",c-Length(nnt)," nonperfect groups");
fi;
Info(InfoLattice,1,Length(nnt)-Length(nt),
" new normal subgroups (",Length(nnt)," total)");
nt:=nnt;
od;
return Reversed(nt); # to stay ascending
end;
#############################################################################
##
#M NormalSubgroups(<G>)
##
InstallMethod(NormalSubgroups,"homomorphism principle pc groups",true,
[IsPcGroup],0,NormalSubgroupsCalc);
InstallMethod(NormalSubgroups,"homomorphism principle perm groups",true,
[IsPermGroup],0,NormalSubgroupsCalc);
#############################################################################
##
#M Socle(<G>)
##
InstallMethod(Socle,"from normal subgroups",true,[IsGroup],0,
function(G)
local n,i,s;
if Size(G)=1 then return G;fi;
# this could be a bit shorter, but the groups in question have few normal
# subgroups
n:=NormalSubgroups(G);
n:=Filtered(n,i->2=Number(n,j->IsSubset(i,j)));
s:=n[1];
for i in [2..Length(n)] do
s:=ClosureGroup(s,n[i]);
od;
return s;
end);
#############################################################################
##
#M IntermediateSubgroups(<G>,<U>)
##
InstallMethod(IntermediateSubgroups,"blocks for coset operation",
IsIdenticalObj, [IsGroup,IsGroup],0,
function(G,U)
local rt,op,a,l,i,j,u,max,subs;
if Length(GeneratorsOfGroup(G))>2 then
a:=SmallGeneratingSet(G);
if Length(a)<Length(GeneratorsOfGroup(G)) then
G:=Subgroup(Parent(G),a);
fi;
fi;
rt:=RightTransversal(G,U);
op:=Action(G,rt,OnRight); # use the special trick for right transversals
a:=ShallowCopy(AllBlocks(op));
l:=Length(a);
# compute inclusion information among sets
Sort(a,function(x,y)return Length(x)<Length(y);end);
# this is n^2 but I hope will not dominate everything.
subs:=List([1..l],i->Filtered([1..i-1],j->IsSubset(a[i],a[j])));
# List the sets we know to be contained in each set
max:=Set(List(Difference([1..l],Union(subs)), # sets which are
# contained in no other
i->[i,l+1]));
for i in [1..l] do
#take all subsets
if Length(subs[i])=0 then
# is minimal
AddSet(max,[0,i]);
else
u:=ShallowCopy(subs[i]);
#and remove those which come via other ones
for j in u do
u:=Difference(u,subs[j]);
od;
for j in u do
#remainder is maximal
AddSet(max,[j,i]);
od;
fi;
od;
return rec(subgroups:=List(a,i->ClosureGroup(U,rt{i})),inclusions:=max);
end);
InstallMethod(IntermediateSubgroups,"normal case",
IsIdenticalObj, [IsGroup,IsGroup],
1,# better than the previous method
function(G,N)
local hom,F,cl,cls,lcl,sub,sel,unsel,i,j;
if not IsNormal(G,N) then
TryNextMethod();
fi;
hom:=NaturalHomomorphismByNormalSubgroup(G,N);
F:=Image(hom,G);
unsel:=[1,Size(F)];
cl:=Filtered(ConjugacyClassesSubgroups(F),
i->not Size(Representative(i)) in unsel);
Sort(cl,function(a,b)
return Size(Representative(a))<Size(Representative(b));
end);
cl:=Concatenation(List(cl,AsList));
lcl:=Length(cl);
cls:=List(cl,Size);
sub:=List(cl,i->[]);
sub[lcl+1]:=[0..Length(cl)];
# now build a list of contained maximal subgroups
for i in [1..lcl] do
sel:=Filtered([1..i-1],j->IsInt(cls[i]/cls[j]) and cls[j]<cls[i]);
# now run through the subgroups in reversed order:
sel:=Reversed(sel);
unsel:=[];
for j in sel do
if not j in unsel then
if IsSubset(cl[i],cl[j]) then
AddSet(sub[i],j);
UniteSet(unsel,sub[j]); # these are not maximal
RemoveSet(sub[lcl+1],j); # j is not maximal in whole
fi;
fi;
od;
if Length(sub[i])=0 then
sub[i]:=[0]; # minimal subgroup
RemoveSet(sub[lcl+1],0);
fi;
od;
sel:=[];
for i in [1..Length(sub)] do
for j in sub[i] do
Add(sel,[j,i]);
od;
od;
return rec(subgroups:=List(cl,i->PreImage(hom,i)),inclusions:=sel);
end);
#############################################################################
##
#E grplatt.gi . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
##
|