User:Tomruen/Finite projective plane configurations

Further information: User:Tomruen/projective_spaces

Finite projective planes

Any finite projective plane of order n, PG(2,n), is an (n²+n+1)_n+1 configuration. Since projective planes are known to exist for all orders n which are powers of primes, these constructions provide infinite families of symmetric configurations.

PG(2,k) exist uniquely for k=2,3,4,5,7,8. For k=9, there are 4 nonisomorphic forms, only one Desarguesian. There are are no Desarguesian solutions for composite k=6,10,12... See Bruck-Ryser-Chowla Theorem.

Automorphisms for PG(2,n), as prime power n=q^m are (m)(n³-1)(n³-n)(n³-n²)/(n-1).^[1]

Reducing projective plane, (n²+n+1)_n+1, by 1 line and its n+1 points becomes [(n²)_n+1 n(n+1)_n]. Automorphisms are reduced by factor n²+n+1. This configuration is an finite affine planes of order n, AP(n). As an affine plane, lines can be partitioned into n+1 sets of n parallel lines and each set can removed independently to create smaller symmetric configurations. Each set of parallel lines pass through every point.

Reducing finite affine plane, [(n²)_n+1 n(n+1)_n], by n lines further becomes a self dual configuration (n²)_n. Automorphisms are reduced by factor n+1. This reduction can be repeated, [(n²)_i ni_n], i=2...n, ending in dual of the complete bipartite graph.

Reducing finite affine plane, [(n²)_n+1 n(n+1)_n], by 1 point and n+1 lines becomes a self dual configuration (n²-1)_n. Automorphisms are reduced by factor n².

Summary

n	Projective plane				Affine plane			Self dual subconfigurations
	Name	Configuration	Aut. group	Order	Name	Configuration	Aut. order	Config	Aut	Config	Aut
2	Fano plane	(73)	PGL(3,2)	168	AG(2,2)	(43 62)	24	(42)	8	(32)	6
3	PG(2,3)	(134)	PGL(3,3)	5,616	AG(2,3)	(94 123)	432	(93)	108	(83)	48
4	PG(2,4)	(215)	PΓL(3,4)	120,960	AG(2,4)	(165 204)	1152	(164)	7560	(154)	360
5	PG(2,5)	(316)	PGL(3,5)	372,000	AG(2,5)	(256 305)	12,000	(255)	2000	(245)	480
6	Does not exist	–	–	–	–	–	–	–	–	–	–
7	PG(2,7)	(578)	PGL(3,7)	5,630,688	AG(2,7)	(498 567)	98,784	(497)	12,348	(487)	2,016
8	PG(2,8)	(739)	PΓL(3,8)	49,448,448	AG(2,8)	(649 728)	677,376	(648)	75,264	(638)	10,584
9	PG(2,9)	(9110)	PΓL(3,9)	84,913,920	AG(2,9)	(8110 909)	933,120	(819)	93,312	(809)	11,520
10	Does not exist	–	–	–	–	–	–	–	–	–	–
11	PG(2,11)	(13312)	PGL(3,11)	212,427,600	AG(2,11)	(12112 13211)	1,597,200	(12111)	133,100	(12011)	13,200

 

Fano plane (7₃) and its reduced configurations, (4₃ 6₂), (4₂) and (3₂)

PG(2,2)

PG(2,2), is the Fano plane, self dual configuration (7₃) has 168 automorphisms = (2³-1)(2³-2)(2³-2²)/(2-1) = 7×6×4

• Reducing 1 line and its 3 points produces the affine plane of order 2, AG(2,2), which the complete graph K₄ (4₃ 6₂), aut(168/7 = 24)
  • Reducing further 2 lines produces complete bipartite graph K_2,2, self dual square (4₂), aut(24/(2+1) = 8).
• Reducing further 1 point and its 3 lines produces complete graph K₃ self dual triangle (3₂), aut(24/2² = 6)

(7₃) is a cyclic configuration with generator line {0,1,3}₇. (4₂), has cyclic generator line {0,1}₄. (3₂) has cyclic generator line {0,1}₃.

PG(2,2)	-1 line, -3 points	-3 line and points	-4 line and points
(73)	(43 62) = K4	(42) = K2,2	(32) = K3
(73)	(43 62)	(42)	(32)
{0,1,3}7		{0,1}4	{0,1}3
0 1 2 3 4 5 6 1 2 3 4 5 6 0 3 4 5 6 0 1 2	0 0 0 1 1 2 1 2 3 2 3 3	0 1 2 3 1 2 3 0	0 1 2 1 2 3

 

PG(2,3), (13₄) and reduced configurations

PG(2,3)

PG(2,3) is self dual configuration (13₄), has aut(5616) = (3³-1)(3³-3)(3³-3²)/(3-1) = 26×24×18/2

• Reducing 1 line and its 4 points produces finite affine plane, AG(2,3), (9₄ 12₃), the Hesse configuration, aut(5616/13 = 432)
  • Reducing further 3 lines produces self dual (9₃), the Pappus configuration, aut(432/(3+1) = 108)
    • Reducing further 3 lines produces (9₂ 6₃), dual of the complete bipartite graph K_3,3, aut(2×(3!)²) = 72.
    • alternately reducing 1 point and its 3 lines makes (6₂+2₂ 6₃), aut(108/3) =36
      • reducing 2 lines and 6 points makes (4₂), aut(8)
  • Reducing further 1 point and its 4 lines produces self dual (8₃), cyclic generator line {0,1,3}₈, the Möbius–Kantor configuration, aut(432/3² = 48)

(13₄) is cyclic configuration, with a generator line as {0,1,4,6}₁₃. (8₃) is cyclic configuration, with a generator line as {0,1,3}₈.

PG(2,4)	(-1 L -4 P)	(-4 L -4 P)	(-7 L -4 P)	(-5 L -5 P)
(134)	(94 123)	(93)	(92 63)	(83)

(13₄) configuration table

0	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	0
4	5	6	7	8	9	10	11	12	0	1	2	3
6	7	8	9	10	11	12	0	1	2	3	4	5

(9₄ 12₃) configuration table

0	3	6	0	1	2	0	1	2	0	1	2
1	4	7	3	4	5	4	5	3	5	3	4
2	5	8	6	7	8	8	6	7	7	8	6

Pappas (9₃) configuration

Set 1			Set 2			Set 3
0	1	4	0	1	2	0	2	3
3	2	5	5	4	3	1	5	4
6	6	6	7	7	7	8	8	8

(8₃) configuration table

0	1	2	3	4	5	6	7
1	2	3	4	5	6	7	0
3	4	5	6	7	0	1	2

 

PG(2,4), (21₅) and reduced configurations

PG(2,4)

PG(2,4) is self dual configuration (21₅) and has aut(120,960) = (2)(4³-1)(4³-4)(4³-4²)/(4-1) = 2×63×60×48/3

• Reducing 1 line and its 5 points produces finite affine plane, AG(2,4), (16₅ 20₄), aut(120960/21 = 5,760)
  • Reducing further 4 lines produces self dual (16₄), aut(5760/(4+1) = 1152)
    • Reducing further 4 lines produces (16₃ 12₄), dual of the Reye configuration, aut(5760/10 = 576)
    • Reducing further 4 lines ends at (16₂ 8₄), dual of complete bipartite graph K_4,4, aut(2×(4!)²) = 1152.
  • Reducing further 1 point and 5 its lines produces self dual (15₄), aut(5760/4² = 360)

(21₅) is a cyclic configuration with generator line {0,3,4,9,11}₂₁.^[2] (15₄) is a cyclic configuration with generating line {0,1,3,7}₁₅.

(21₅) configuration table

0	1	2	3	4	5	6	7	8	9	10	11	12
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
9	10	11	12	13	14	15	16	17	18	19	20	0
11	12	13	14	15	16	17	18	19	20	0	1	2

(16₅ 20₄) configuration table

1	2	13	14	12	13	14	15	3	16	5	3	4	5	6	7	8	9	10	11
12	12	14	15	15	16	16	3	4	4	6	6	7	8	9	10	11	1	1	2
16	13	3	4	5	6	7	8	9	5	11	7	8	9	10	11	12	13	2	15
3	4	5	6	7	8	9	10	11	10	1	2	1	2	12	13	14	15	14	16

(15₄) configuration table

0	1	2	3	4	5	6	7	8	9	10	11	0	1	2
1	2	3	4	5	6	7	8	9	10	11	0	1	2	3
3	4	5	6	7	8	9	10	11	0	1	2	3	4	5
7	8	9	10	11	0	1	2	3	4	5	6	7	8	9

 

PG(2,5), (31₆) and reduced configurations

PG(2,5)

PG(2,5) is self dual configuration is (31₆) and has aut(372,000) = (5³-1)(5³-5)(5³-5²)/(5-1) = 124×120×100/4

• Reducing 1 line and its 6 points produces finite affine plane AG(2,5), configuration (25₆ 30₅) aut(372000/31 = 12,000)
  • Reducing further 5 lines produces self dual (25₅), aut(12000/(5+1) = 2000)
    • Reducing further multiples of 5 lines produces (25₄ 20₅), (25₃ 15₅) and ending at (25₂ 10₅), dual of complete bipartite graph K_5,5, aut(2×(5!)²) = 28,800.
  • Reducing further 1 point and its 6 lines produces self dual (24₅), aut(12000/5² = 480)

(31₆) is a cyclic configuration with generator line {0,1,3,8,12,18}₃₁. (24₅) is a cyclic configuration with generator line {0,1,4,9,11}₂₄.

(31₆) configuration table

0	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
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	0
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	0	1	2
8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	0	1	2	3	4	5	6	7
12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	0	1	2	3	4	5	6	7	8	9	10	11
18	19	20	21	22	23	24	25	26	27	28	29	30	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17

(24₅) configuration table

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	0
4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	0	1	2	3
9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	0	1	2	3	4	5	6	7	8
11	12	13	14	15	16	17	18	19	20	21	22	23	0	1	2	3	4	5	6	7	8	9	10

PG(2,6)

PG(2,6) would be (43₇), has no cyclic generators and the finite geometry doesn't exist as 6 is composite.

PG(2,7)

PG(2,7) is self dual configuration (57₈) and has aut(5,630,688) = (7³-1)(7³-7)(7³-7²)/(7-1) = 342×336×294/6

• Reducing 1 line and its 8 points produces finite affine plane AG(2,7), configuration (49₈ 56₇), aut(5630688/57 = 98,784)
  • Reducing further 7 lines produces self dual (49₇), aut(98784/(7+1) = 12,348)
    • Reducing further multiples 7 lines produces (49₆ 42₇), (49₅ 35₇), (49₄ 28₇), (49₃ 21₇) and ending at (49₂ 14₇), dual of complete bipartite graph K_7,7, aut(2×(7!)²) = 50,803,200.
  • Reducing further 1 point and its 8 lines produces self dual (48₇), aut(98784/7² = 2016)

(57₈) is a cyclic configuration with generator line {0,1,3,13,32,36,43,52}₅₇. (48₇) is a cyclic configuration with generator line {0, 1, 3, 15, 20, 38, 42}₄₈.

(57₈) configuration table

0	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
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	0
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	0	1	2
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	0	1	2	3	4	5	6	7	8	9	10	11	12
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	0	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
36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	0	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
43	44	45	46	47	48	49	50	51	52	53	54	55	56	0	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
52	53	54	55	56	0	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

PG(2,8)

PG(2,8) is self dual configuration (73₉) has aut(49,448,448) = (3)(8³-1)(8³-8)(8³-8²)/(8-1) = 3×511×504×448/7

• Reducing 1 line and its 9 points produces finite affine plane AG(2,8), configuration (64₉ 72₈), aut(49448448/73 = 677,376)
  • Reducing further 8 lines produces self dual (64₈), aut(677376/(8+1) = 75,264)
    • Reducing further multiples of 8 lines produces (64₇ 56₈), (64₆ 48₈), (64₅ 40₈), (64₄ 32₈), (64₃ 24₈), and ending at (64₂ 16₈), dual of complete bipartite graph K_8,8, aut(2×(8!)²) = 3,251,404,800.
  • Reducing further 1 point and its 9 lines produces self dual (63₈), aut(677376/8² = 10,584)

(73₉) is a cyclic configuration with generator line {0,1,3,7,15,31,36,54,63}₇₃.

(73₉) configuration table

0	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
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	0
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	0	1	2
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	0	1	2	3	4	5	6
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	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
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	0	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
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	0	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
54	55	56	57	58	59	60	61	62	63	64	65	66	67	68	69	70	71	72	0	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
63	64	65	66	67	68	69	70	71	72	0	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

PG(2,9)

PG(2,9) is a self dual configuration (91₁₀) with 4 unique nonisomorphic solutions, only one Desarguesian, aut(84,913,920) = 2(9³-1)(9³-9)(9³-9²)/(9-1) = 2×728×720×648/8

• Reducing 1 line and its 9 points produces finite affine plane AG(2,9), configuration (81₁₀ 90₉), aut(84913920/91 = 933,120)
  • Reducing further 8 lines produces self dual (81₉), aut(933120/(9+1) = 93,312)
    • Reducing further multiples of 8 lines produces (81₈ 72₉), (64₇ 64₉), (81₆ 56₉), (81₅ 45₉), (81₄ 36₉), (81₃ 27₉), and ending at (81₂ 18₉), dual of complete bipartite graph K_9,9, aut(2×(9!)²) = 263,363,788,800.
  • Reducing further 1 point and its 9 lines produces self dual (80₉), aut(933120/9² = 11,520)

It has cyclic generator {0,1,3,9,27,49,56,61,77,81}₉₁.

PG(2,10)

PG(2,10) would be (111₁₁), but as 10 is composite, there are no solutions.

PG(2,11)

PG(2,11) is self dual configuration (133₁₂). Aut(212,427,600) = (11³-1)(11³-11)(11³-11²)/(11-1) = 1330×1320×1210/10

Notes

1. Finite Geometries, by Peter Dembowski (1968)
2. Grünbaum (2009), p.234

References

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