813:, meeting the translation line in 4 points. The latter of these two unitals was shown by Grüning to also be embeddable in the dual Hall plane. Another of the unitals arises from the construction of Barlotti and Lunardon. The fourth has an automorphism group of order 8 isomorphic to the
704:
in 1907. There are four quasifields of order nine which can be used to construct the Hall plane of order nine. Three of these are Hall systems generated by the irreducible polynomials
426:
A process, due to T. G. Ostrom, which replaces certain sets of lines in a projective plane by alternate sets in such a way that the new structure is still a projective plane is called
1403:
1304:
927:
1445:
761:, and it was in this context that the plane was discovered by Veblen and Wedderburn. This plane is often referred to as the nearfield plane of order nine.
1023:
1296:
1378:
1359:
996:
972:
21:
1395:
919:
1292:
681:
582:
25:
1468:
696:
While usually constructed in the same way as other Hall planes, the Hall plane of order 9 was actually found earlier by
786:
1473:
758:
775:
677:
627:
844:
Although the constructions will provide a projective plane of order 4, the unique such plane is
1422:
1331:
1241:
1186:
1131:
1076:
946:
845:
779:
420:
1391:
1374:
1355:
1323:
1233:
1178:
1123:
1068:
1029:
1019:
992:
968:
915:
802:
789:, having 5 pairs of points that the group preserves set-wise; the automorphism group acts as S
783:
701:
664:
572:
75:
a prime. The creation of the plane from the quasifield follows the standard construction (see
1412:
1313:
1225:
1170:
1115:
1060:
936:
491:
450:
442:
431:
1343:
1339:
774:
The Hall plane of order 9 is the unique projective plane, finite or infinite, which has
1433:
599:
1437:
1462:
1387:
1245:
1190:
1080:
911:
697:
419:
Another construction that produces Hall planes is obtained by applying derivation to
1135:
685:
83:
295:. The properties of the defined multiplication which turn the right vector space
676:
is the smallest Hall plane, and one of the three smallest examples of a finite
1150:
814:
589:
76:
56:
1327:
1237:
1182:
1127:
1072:
1033:
1013:
1259:
Barlotti, A.; Lunardon, G. (1979). "Una classe di unitals nei Δ-piani".
895:
1426:
1335:
1229:
1174:
1119:
1064:
950:
1205:
1095:
1048:
809:, meeting the translation line in a single point, while the other is
805:. Two of these unitals arise from Buekenhout's constructions: one is
1417:
1318:
941:
757:. The first of these produces an associative quasifield, that is, a
595:
All finite Hall planes contain subplanes of order different from 2.
145:, to a quasifield by defining a multiplication on the vectors by
1018:. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 126.
55:
The original construction of Hall planes was based on the Hall
967:, San Francisco: W.H. Freeman and Company, pp. 333–334,
1151:"Existence of unitals in finite translation planes of order
801:
The Hall plane of order 9 admits four inequivalent embedded
901:
explicitly lists the incidence structure of these planes.
578:
All finite Hall planes of the same order are isomorphic.
1210:
which can be embedded in two different planes of order
588:
All finite Hall planes contain subplanes of order 2 (
406:
commutes (multiplicatively) with all the elements of
1104:) in the affine and projective planes of order nine"
1015:
430:. We give the details of this process. Start with a
848:
and is generally not considered to be a Hall plane.
660:
652:
644:
636:
626:
618:
613:
558:and it, or its projective completion, is called a
1404:Transactions of the American Mathematical Society
1305:Transactions of the American Mathematical Society
928:Transactions of the American Mathematical Society
1261:Rivisita di Matematica della Università di Parma
817:, and is not part of any known infinite family.
1396:"Non-Desarguesian and non-Pascalian geometries"
920:"Non-Desarguesian and non-Pascalian geometries"
486:which determine a line meeting ℓ in a point of
101:a prime and a quadratic irreducible polynomial
1094:Penttila, Tim; Royle, Gordon F. (1995-11-01).
8:
1446:Notices of the American Mathematical Society
1373:, San Francisco: W.H. Freeman and Company,
881:
869:
857:
872:, pp. 202–218, Chapter X. Derivation)
1416:
1317:
940:
82:To build a Hall quasifield, start with a
542:) and the Baer subplanes that belong to
826:
51:Algebraic construction via Hall systems
610:
141:, a two-dimensional vector space over
474:if for every pair of distinct points
7:
1049:"Projektive Ebenen über Fastkörpern"
833:
28:(1943). There are examples of order
1438:"Survey of Non-Desarguesian Planes"
534:which do not meet ℓ at a point of
272:, we can identify the elements of
14:
782:acts on its (necessarily unique)
678:non-Desarguesian projective plane
506:(we say that such Baer subplanes
315:satisfies the quadratic equation
22:non-Desarguesian projective plane
1369:Stevenson, Frederick W. (1972),
991:. Springer-Verlag. p. 186.
963:Stevenson, Frederick W. (1972),
897:Projective Planes of Small Order
441:and designate one line ℓ as its
1108:Designs, Codes and Cryptography
987:D. Hughes and F. Piper (1973).
514:.) Define a new affine plane D(
1350:Hughes, D.; Piper, F. (1973).
1047:André, Johannes (1955-12-01).
554:) is an affine plane of order
518:) as follows: The points of D(
1:
1206:"A class of unitals of order
1204:Grüning, Klaus (1987-06-01).
1149:Buekenhout, F. (July 1976).
884:, p. 203, Theorem 10.2)
894:Moorhouse, G. Eric (2017),
36:and every positive integer
1490:
1287:, Berlin: Springer-Verlag
1053:Mathematische Zeitschrift
1012:Dembowski, Peter (1968).
882:Hughes & Piper (1973
870:Hughes & Piper (1973
858:Hughes & Piper (1973
470:points of ℓ is called a
226:Writing the elements of
1392:Wedderburn, Joseph H.M.
1155:with a kernel of order
916:Wedderburn, Joseph H.M.
299:into a quasifield are:
238:, that is, identifying
1283:Dembowski, P. (1968),
674:Hall plane of order 9
614:Hall plane of order 9
607:Hall plane of order 9
276:as the ordered pairs
581:Hall planes are not
522:) are the points of
332:is in the kernel of
230:in terms of a basis
1469:Projective geometry
1354:. Springer-Verlag.
1297:"Projective Planes"
1218:Journal of Geometry
1163:Geometriae Dedicata
776:Lenz–Barlotti class
645:Point orbit lengths
628:Lenz–Barlotti class
530:) are the lines of
421:Desarguesian planes
1293:Hall, Marshall Jr.
1230:10.1007/BF01234988
1175:10.1007/BF00145956
1120:10.1007/BF01388477
1065:10.1007/BF01180628
793:on these 5 pairs.
780:automorphism group
770:Automorphism Group
653:Line orbit lengths
573:translation planes
16:In mathematics, a
1371:Projective Planes
1352:Projective Planes
1285:Finite Geometries
1025:978-3-642-62012-6
989:Projective Planes
965:Projective Planes
702:Joseph Wedderburn
680:, along with its
670:
669:
665:Translation plane
526:. The lines of D(
402:every element of
26:Marshall Hall Jr.
1481:
1454:
1442:
1429:
1420:
1400:
1383:
1365:
1346:
1321:
1301:
1288:
1269:
1268:
1256:
1250:
1249:
1201:
1195:
1194:
1146:
1140:
1139:
1091:
1085:
1084:
1044:
1038:
1037:
1009:
1003:
1002:
984:
978:
977:
960:
954:
953:
944:
924:
908:
902:
900:
891:
885:
879:
873:
867:
861:
855:
849:
842:
836:
831:
784:translation line
756:
737:
718:
611:
598:Hall planes are
571:Hall planes are
469:
458:
443:line at infinity
432:projective plane
378:
358:
325:
294:
283:
259:
249:
237:
222:
198:
191:
140:
122:
96:
46:
32:for every prime
1489:
1488:
1484:
1483:
1482:
1480:
1479:
1478:
1474:Finite geometry
1459:
1458:
1457:
1453:(10): 1294–1303
1440:
1434:Weibel, Charles
1432:
1418:10.2307/1988781
1398:
1386:
1381:
1368:
1362:
1349:
1319:10.2307/1990331
1299:
1291:
1282:
1278:
1273:
1272:
1258:
1257:
1253:
1203:
1202:
1198:
1148:
1147:
1143:
1096:"Sets of type (
1093:
1092:
1088:
1046:
1045:
1041:
1026:
1011:
1010:
1006:
999:
986:
985:
981:
975:
962:
961:
957:
942:10.2307/1988781
922:
910:
909:
905:
893:
892:
888:
880:
876:
868:
864:
856:
852:
843:
839:
832:
828:
823:
799:
792:
772:
767:
739:
720:
705:
694:
609:
568:
546:(restricted to
538:(restricted to
464:
453:
417:
360:
337:
316:
285:
277:
251:
239:
231:
200:
193:
146:
128:
102:
87:
59:(also called a
53:
41:
24:constructed by
12:
11:
5:
1487:
1485:
1477:
1476:
1471:
1461:
1460:
1456:
1455:
1430:
1411:(3): 379–388,
1388:Veblen, Oswald
1384:
1379:
1366:
1360:
1347:
1312:(2): 229–277,
1289:
1279:
1277:
1274:
1271:
1270:
1251:
1196:
1141:
1114:(3): 229–245.
1086:
1059:(1): 137–160.
1039:
1024:
1004:
997:
979:
973:
955:
935:(3): 379–388,
912:Veblen, Oswald
903:
886:
874:
862:
860:, p. 183)
850:
837:
825:
824:
822:
819:
798:
795:
790:
771:
768:
766:
763:
693:
690:
668:
667:
662:
658:
657:
654:
650:
649:
646:
642:
641:
638:
634:
633:
630:
624:
623:
620:
616:
615:
608:
605:
604:
603:
596:
593:
590:Fano subplanes
586:
579:
576:
567:
564:
472:derivation set
416:
413:
412:
411:
400:
336:(meaning that
327:
303:every element
79:for details).
52:
49:
13:
10:
9:
6:
4:
3:
2:
1486:
1475:
1472:
1470:
1467:
1466:
1464:
1452:
1448:
1447:
1439:
1435:
1431:
1428:
1424:
1419:
1414:
1410:
1406:
1405:
1397:
1393:
1389:
1385:
1382:
1380:0-7167-0443-9
1376:
1372:
1367:
1363:
1361:0-387-90044-6
1357:
1353:
1348:
1345:
1341:
1337:
1333:
1329:
1325:
1320:
1315:
1311:
1307:
1306:
1298:
1294:
1290:
1286:
1281:
1280:
1275:
1266:
1262:
1255:
1252:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1213:
1209:
1200:
1197:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1158:
1154:
1145:
1142:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1103:
1099:
1090:
1087:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1055:(in German).
1054:
1050:
1043:
1040:
1035:
1031:
1027:
1021:
1017:
1016:
1008:
1005:
1000:
998:0-387-90044-6
994:
990:
983:
980:
976:
974:0-7167-0443-9
970:
966:
959:
956:
952:
948:
943:
938:
934:
930:
929:
921:
917:
913:
907:
904:
899:
898:
890:
887:
883:
878:
875:
871:
866:
863:
859:
854:
851:
847:
841:
838:
835:
830:
827:
820:
818:
816:
812:
808:
804:
796:
794:
788:
787:imprimitively
785:
781:
777:
769:
764:
762:
760:
754:
750:
746:
742:
735:
731:
727:
723:
716:
712:
708:
703:
699:
698:Oswald Veblen
691:
689:
687:
683:
679:
675:
666:
663:
659:
655:
651:
647:
643:
639:
637:Automorphisms
635:
631:
629:
625:
621:
617:
612:
606:
601:
597:
594:
591:
587:
584:
580:
577:
574:
570:
569:
565:
563:
561:
560:derived plane
557:
553:
550:). The set D(
549:
545:
541:
537:
533:
529:
525:
521:
517:
513:
509:
505:
501:
497:
493:
492:Baer subplane
490:, there is a
489:
485:
481:
477:
473:
467:
462:
456:
452:
448:
444:
440:
436:
433:
429:
424:
422:
414:
409:
405:
401:
398:
394:
390:
386:
382:
376:
372:
368:
364:
357:
353:
349:
345:
341:
335:
331:
328:
323:
319:
314:
310:
306:
302:
301:
300:
298:
292:
288:
281:
275:
271:
267:
263:
258:
254:
247:
243:
235:
229:
224:
220:
216:
212:
208:
204:
196:
189:
185:
181:
177:
173:
170:
166:
162:
158:
154:
150:
144:
139:
135:
131:
126:
121:
117:
113:
109:
105:
100:
94:
90:
85:
80:
78:
74:
70:
66:
62:
58:
50:
48:
44:
39:
35:
31:
27:
23:
19:
1450:
1444:
1408:
1402:
1370:
1351:
1309:
1303:
1284:
1264:
1260:
1254:
1224:(1): 61–77.
1221:
1217:
1211:
1207:
1199:
1166:
1162:
1156:
1152:
1144:
1111:
1107:
1101:
1097:
1089:
1056:
1052:
1042:
1014:
1007:
988:
982:
964:
958:
932:
926:
906:
896:
889:
877:
865:
853:
846:Desarguesian
840:
829:
810:
806:
800:
773:
752:
748:
744:
740:
733:
729:
725:
721:
714:
710:
706:
695:
692:Construction
688:of order 9.
686:Hughes plane
673:
671:
600:André planes
559:
555:
551:
547:
543:
539:
535:
531:
527:
523:
519:
515:
511:
507:
503:
499:
495:
487:
483:
479:
475:
471:
465:
460:
454:
451:affine plane
446:
438:
434:
427:
425:
418:
407:
403:
396:
392:
388:
384:
380:
374:
370:
366:
362:
355:
351:
347:
343:
339:
333:
329:
321:
317:
312:
308:
304:
296:
290:
286:
279:
273:
269:
265:
261:
256:
252:
245:
241:
233:
227:
225:
218:
214:
210:
206:
202:
194:
187:
183:
179:
175:
171:
168:
164:
160:
156:
152:
148:
142:
137:
133:
129:
124:
119:
115:
111:
107:
103:
98:
92:
88:
84:Galois field
81:
72:
68:
64:
60:
54:
42:
37:
33:
29:
17:
15:
834:Hall (1943)
815:quaternions
778:IVa.3. Its
494:containing
232:⟨1,
223:otherwise.
61:Hall system
1463:Categories
1276:References
1267:: 781–785.
811:hyperbolic
765:Properties
759:near-field
661:Properties
566:Properties
428:derivation
415:Derivation
268:vary over
77:quasifield
57:quasifield
18:Hall plane
1328:0002-9947
1246:117872040
1238:1420-8997
1191:123037502
1183:0046-5755
1128:1573-7586
1081:122641224
1073:1432-1823
1034:851794158
807:parabolic
640:2 × 3 × 5
583:self-dual
457:∖ ℓ
437:of order
127:. Extend
67:of order
40:provided
1436:(2007),
1394:(1907),
1295:(1943),
1136:43638589
918:(1907),
684:and the
459:. A set
391:and all
379:for all
236:⟩
213:, 0) = (
1427:1988781
1344:0008892
1336:1990331
951:1988781
803:unitals
797:Unitals
449:be the
311:not in
284:, i.e.
1425:
1377:
1358:
1342:
1334:
1326:
1244:
1236:
1189:
1181:
1134:
1126:
1079:
1071:
1032:
1022:
995:
971:
949:
648:10, 81
508:belong
445:. Let
399:); and
359:, and
45:> 4
1441:(PDF)
1423:JSTOR
1399:(PDF)
1332:JSTOR
1300:(PDF)
1242:S2CID
1187:S2CID
1169:(2).
1132:S2CID
1077:S2CID
947:JSTOR
923:(PDF)
821:Notes
656:1, 90
632:IVa.3
619:Order
324:) = 0
250:with
209:) ∘ (
192:when
163:) = (
155:) ∘ (
123:over
91:= GF(
20:is a
1375:ISBN
1356:ISBN
1324:ISSN
1234:ISSN
1179:ISSN
1124:ISSN
1069:ISSN
1030:OCLC
1020:ISBN
993:ISBN
969:ISBN
747:) =
728:) =
713:) =
700:and
682:dual
672:The
502:and
478:and
282:, 0)
264:and
199:and
110:) =
97:for
71:for
1413:doi
1314:doi
1226:doi
1171:doi
1116:doi
1061:doi
937:doi
755:− 1
738:or
736:− 1
717:+ 1
510:to
482:of
468:+ 1
463:of
395:in
387:in
307:of
260:as
197:≠ 0
178:),
63:),
1465::
1451:54
1449:,
1443:,
1421:,
1407:,
1401:,
1390:;
1340:MR
1338:,
1330:,
1322:,
1310:54
1308:,
1302:,
1263:.
1240:.
1232:.
1222:29
1220:.
1216:.
1185:.
1177:.
1165:.
1161:.
1130:.
1122:.
1110:.
1106:.
1100:,
1075:.
1067:.
1057:62
1051:.
1028:.
945:,
931:,
925:,
914:;
751:+
732:−
719:,
592:).
562:.
498:,
423:.
383:,
375:βc
369:=
363:αβ
356:βc
354:+
352:αc
350:=
342:+
289:+
257:λy
255:+
244:,
219:bc
217:,
215:ac
205:,
188:br
186:+
184:bc
182:−
180:ad
169:bd
167:−
165:ac
159:,
151:,
136:×
132:=
118:−
116:rx
114:−
86:,
47:.
1415::
1409:8
1364:.
1316::
1265:4
1248:.
1228::
1214:"
1212:q
1208:q
1193:.
1173::
1167:5
1159:"
1157:q
1153:q
1138:.
1118::
1112:6
1102:n
1098:m
1083:.
1063::
1036:.
1001:.
939::
933:8
791:5
753:x
749:x
745:x
743:(
741:h
734:x
730:x
726:x
724:(
722:g
715:x
711:x
709:(
707:f
622:9
602:.
585:.
575:.
556:n
552:A
548:A
544:D
540:A
536:D
532:π
528:A
524:A
520:A
516:A
512:D
504:D
500:Y
496:X
488:D
484:A
480:Y
476:X
466:n
461:D
455:π
447:A
439:n
435:π
410:.
408:H
404:F
397:F
393:c
389:H
385:β
381:α
377:)
373:(
371:α
367:c
365:)
361:(
348:c
346:)
344:β
340:α
338:(
334:H
330:F
326:;
322:α
320:(
318:f
313:F
309:H
305:α
297:H
293:0
291:λ
287:x
280:x
278:(
274:F
270:F
266:y
262:x
253:x
248:)
246:y
242:x
240:(
234:λ
228:H
221:)
211:c
207:b
203:a
201:(
195:d
190:)
176:c
174:(
172:f
161:d
157:c
153:b
149:a
147:(
143:F
138:F
134:F
130:H
125:F
120:s
112:x
108:x
106:(
104:f
99:p
95:)
93:p
89:F
73:p
69:p
65:H
43:p
38:n
34:p
30:p
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.