check_note.py -- Kourovka Problem 7.27 (corpus record KOU-7.27)
Part A: field self-test (all prime powers q <= 256) and group orders in PGL_2(q)
  fields built and tested: 70
  group orders by the stabiliser chain = brute-force closure: 700 pairs
Part B: Lemma 3.3(a): |T \ R_N| <= n(n-1) N^2 |T|/(q-1), N = n!
  n=2, 197 prime powers q <= 1024: bound holds for all; |R_N|/|T| for some q: q=16: 0.9333, q=29: 0.8571, q=59: 0.9655, q=64: 0.9841, q=83: 0.9756, q=101: 0.9600, q=107: 0.9811, q=127: 0.9841, q=128: 0.9921, q=179: 0.9888, q=227: 0.9912, q=251: 0.9920, q=256: 0.9961, q=512: 0.9980, q=1024: 0.9990
  n=3, 69 prime powers q <= 256: bound holds for all; |R_N|/|T| for some q: q=16: 0.3200, q=29: 0.6122, q=59: 0.8989, q=64: 0.8707, q=83: 0.9280, q=101: 0.7752, q=107: 0.9441, q=127: 0.8707, q=128: 0.9765, q=179: 0.9665, q=227: 0.9736, q=251: 0.9293, q=256: 0.9201
  n=4, 43 prime powers q <= 128: bound holds for all; |R_N|/|T| for some q: q=16: 0.0000, q=29: 0.0000, q=59: 0.8059, q=64: 0.0000, q=83: 0.8601, q=101: 0.1920, q=107: 0.8907, q=127: 0.0691, q=128: 0.9534
Part C: Lemma 3.3(b): class of a regular diagonal d meets T in n! elements; C(d) = T
  n=2, q=4: 2 regular diagonal elements checked
  n=2, q=5: 2 regular diagonal elements checked
  n=2, q=7: 4 regular diagonal elements checked
  n=2, q=8: 6 regular diagonal elements checked
  n=2, q=9: 6 regular diagonal elements checked
  n=2, q=11: 8 regular diagonal elements checked
  n=2, q=13: 10 regular diagonal elements checked
  n=3, q=4: |SL_3(4)| = 60480, |C(d)| = 9 = |T|, class meets T in 6 elements (0.3 s)
  n=3, q=5: |SL_3(5)| = 372000, |C(d)| = 16 = |T|, class meets T in 6 elements (2.0 s)
Part D: Table 2 (n = 2, 4 <= q <= 64): P(d) by Proposition 6.3 (Dickson) and by direct generation (Lemma 6.4)
  q= 4: P(d) holds for ord(l) in [3]; fails for 1 [|H/Z|=5; m<=2: nonsplit torus or its normaliser]  (0.0 s)
  q= 5: P(d) holds for ord(l) in []; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 4 [|H/Z|=12; m<=2: nonsplit torus or its normaliser]  (0.0 s)
  q= 7: P(d) holds for ord(l) in []; fails for 1 [|H/Z|=4; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=4; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=24; m=3: A_4], 6 [|H/Z|=24; m=3: A_4]  (0.0 s)
  q= 8: P(d) holds for ord(l) in [7]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser]  (0.0 s)
  q= 9: P(d) holds for ord(l) in []; fails for 1 [|H/Z|=5; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=5; m<=2: nonsplit torus or its normaliser], 4 [|H/Z|=24; m<=2: nonsplit torus or its normaliser], 8 [|H/Z|=24; m=4: S_4]  (0.0 s)
  q=11: P(d) holds for ord(l) in []; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 5 [|H/Z|=60; m=5: A_5], 10 [|H/Z|=60; m=5: A_5]  (0.0 s)
  q=13: P(d) holds for ord(l) in [12]; fails for 1 [|H/Z|=7; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=7; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=12; m=3: A_4], 4 [|H/Z|=14; m<=2: nonsplit torus or its normaliser], 6 [|H/Z|=12; m=3: A_4]  (0.0 s)
  q=16: P(d) holds for ord(l) in [15]; fails for 1 [|H/Z|=17; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=60; m=3: A_5 = PSL_2(4)], 5 [|H/Z|=60; m=5: A_5]  (0.0 s)
  q=17: P(d) holds for ord(l) in [16]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 4 [|H/Z|=24; m<=2: nonsplit torus or its normaliser], 8 [|H/Z|=24; m=4: S_4]  (0.0 s)
  q=19: P(d) holds for ord(l) in [9, 18]; fails for 1 [|H/Z|=5; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=5; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=60; m=3: A_4], 6 [|H/Z|=60; m=3: A_4]  (0.0 s)
  q=23: P(d) holds for ord(l) in [11, 22]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser]  (0.0 s)
  q=25: P(d) holds for ord(l) in [24]; fails for 1 [|H/Z|=13; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=13; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=24; m=3: A_4], 4 [|H/Z|=60; m<=2: nonsplit torus or its normaliser], 6 [|H/Z|=24; m=3: A_4], 8 [|H/Z|=120; m=4: S_4], 12 [|H/Z|=120; m=6: PGL_2(5)]  (0.0 s)
  q=27: P(d) holds for ord(l) in [13, 26]; fails for 1 [|H/Z|=14; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=14; m<=2: nonsplit torus or its normaliser]  (0.0 s)
  q=29: P(d) holds for ord(l) in [7, 14, 28]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 4 [|H/Z|=10; m<=2: nonsplit torus or its normaliser]  (0.0 s)
  q=31: P(d) holds for ord(l) in [15, 30]; fails for 1 [|H/Z|=16; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=16; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=60; m=3: A_4], 5 [|H/Z|=60; m=5: A_5], 6 [|H/Z|=60; m=3: A_4], 10 [|H/Z|=60; m=5: A_5]  (0.0 s)
  q=32: P(d) holds for ord(l) in [31]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser]  (0.0 s)
  q=37: P(d) holds for ord(l) in [9, 12, 18, 36]; fails for 1 [|H/Z|=19; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=19; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=12; m=3: A_4], 4 [|H/Z|=38; m<=2: nonsplit torus or its normaliser], 6 [|H/Z|=12; m=3: A_4]  (0.1 s)
  q=41: P(d) holds for ord(l) in [20, 40]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 4 [|H/Z|=60; m<=2: nonsplit torus or its normaliser], 5 [|H/Z|=60; m=5: A_5], 8 [|H/Z|=24; m=4: S_4], 10 [|H/Z|=60; m=5: A_5]  (0.1 s)
  q=43: P(d) holds for ord(l) in [7, 14, 21, 42]; fails for 1 [|H/Z|=22; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=22; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=12; m=3: A_4], 6 [|H/Z|=12; m=3: A_4]  (0.2 s)
  q=47: P(d) holds for ord(l) in [23, 46]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser]  (0.0 s)
  q=49: P(d) holds for ord(l) in [24, 48]; fails for 1 [|H/Z|=25; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=25; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=168; m=3: A_4], 4 [|H/Z|=336; m<=2: nonsplit torus or its normaliser], 6 [|H/Z|=168; m=3: A_4], 8 [|H/Z|=336; m=4: S_4], 12 [|H/Z|=336; m=6: PGL_2(7)], 16 [|H/Z|=336; m=8: PGL_2(7)]  (0.0 s)
  q=53: P(d) holds for ord(l) in [13, 26, 52]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 4 [|H/Z|=54; m<=2: nonsplit torus or its normaliser]  (0.1 s)
  q=59: P(d) holds for ord(l) in [29, 58]; fails for 1 [|H/Z|=3; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=3; m<=2: nonsplit torus or its normaliser]  (0.1 s)
  q=61: P(d) holds for ord(l) in [12, 15, 20, 30, 60]; fails for 1 [|H/Z|=31; m<=2: nonsplit torus or its normaliser], 2 [|H/Z|=31; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=60; m=3: A_4], 4 [|H/Z|=60; m<=2: nonsplit torus or its normaliser], 5 [|H/Z|=60; m=5: A_5], 6 [|H/Z|=60; m=3: A_4], 10 [|H/Z|=60; m=5: A_5]  (0.5 s)
  q=64: P(d) holds for ord(l) in [21, 63]; fails for 1 [|H/Z|=65; m<=2: nonsplit torus or its normaliser], 3 [|H/Z|=60; m=3: A_5 = PSL_2(4)], 7 [|H/Z|=504; m=7: PSL_2(8)], 9 [|H/Z|=504; m=9: PSL_2(8)]  (0.1 s)
  64 < q <= 128: Proposition 6.3 and direct generation agree for all 107 pairs (q, ord(l)), ord(l) not dividing 4, 17 prime powers (35.5 s)
Part E: Table 1 (n = 2, q in {2,3,5,7,9,11}): certified witnesses for every diagonal d
  (H proper, d in H, no orbit of H on P^1(F_q) with fewer than 3 points; 'nonabelian' means absolutely irreducible)
  q=2: ord(d)=1: |H|=3 abelian, orbits [3], H=<d,(0, 1, 1, 1)>
  q=3: ord(d)=1: |H|=8 nonabelian, orbits [4], H=<d,(0, 1, 2, 0),(1, 1, 1, 2)>; ord(d)=2: |H|=8 nonabelian, orbits [4], H=an earlier witness
  q=5: ord(d)=4: |H|=24 nonabelian, orbits [6], H=<d,(1, 1, 2, 3)>; ord(d)=4: |H|=24 nonabelian, orbits [6], H=<d,(1, 1, 2, 3)>; ord(d)=1: |H|=24 nonabelian, orbits [6], H=an earlier witness; ord(d)=2: |H|=24 nonabelian, orbits [6], H=an earlier witness
  q=7: ord(d)=3: |H|=48 nonabelian, orbits [8], H=<d,(1, 1, 2, 3)>; ord(d)=6: |H|=48 nonabelian, orbits [8], H=<d,(1, 1, 2, 3)>; ord(d)=3: |H|=48 nonabelian, orbits [8], H=<d,(1, 1, 2, 3)>; ord(d)=6: |H|=48 nonabelian, orbits [8], H=<d,(1, 1, 2, 3)>; ord(d)=1: |H|=48 nonabelian, orbits [8], H=an earlier witness; ord(d)=2: |H|=48 nonabelian, orbits [8], H=an earlier witness
  q=9: ord(d)=8: |H|=48 nonabelian, orbits [4, 6], H=<d,(1, 1, 1, 2)>; ord(d)=8: |H|=48 nonabelian, orbits [4, 6], H=<d,(1, 1, 1, 2)>; ord(d)=4: |H|=48 nonabelian, orbits [4, 6], H=<d,(0, 1, 2, 1)>; ord(d)=8: |H|=48 nonabelian, orbits [4, 6], H=<d,(1, 1, 1, 2)>; ord(d)=4: |H|=48 nonabelian, orbits [4, 6], H=<d,(0, 1, 2, 1)>; ord(d)=8: |H|=48 nonabelian, orbits [4, 6], H=<d,(1, 1, 1, 2)>; ord(d)=1: |H|=48 nonabelian, orbits [4, 6], H=an earlier witness; ord(d)=2: |H|=48 nonabelian, orbits [4, 6], H=an earlier witness
  q=11: ord(d)=10: |H|=120 nonabelian, orbits [12], H=<d,(1, 1, 1, 2)>; ord(d)=5: |H|=120 nonabelian, orbits [12], H=<d,(1, 1, 1, 2)>; ord(d)=5: |H|=120 nonabelian, orbits [12], H=<d,(1, 1, 1, 2)>; ord(d)=5: |H|=120 nonabelian, orbits [12], H=<d,(1, 1, 1, 2)>; ord(d)=10: |H|=120 nonabelian, orbits [12], H=<d,(1, 1, 1, 2)>; ord(d)=10: |H|=120 nonabelian, orbits [12], H=<d,(1, 1, 1, 2)>; ord(d)=10: |H|=120 nonabelian, orbits [12], H=<d,(1, 1, 1, 2)>; ord(d)=5: |H|=120 nonabelian, orbits [12], H=<d,(1, 1, 1, 2)>; ord(d)=1: |H|=120 nonabelian, orbits [12], H=an earlier witness; ord(d)=2: |H|=120 nonabelian, orbits [12], H=an earlier witness
  q=2: every proper subgroup of SL_2(2) (all generated by two elements) is abelian: True
Part F: Theorem 1.2, d = diag(w, 1/w) with w primitive
  direct generation: P holds for all 62 prime powers 13 <= q <= 256 (28.6 s)
  arithmetic of Theorem 1.2: m = (q-1)/gcd(2,q-1) >= 6 and m > q0 + 1 for every proper subfield F_q0, for all 2320 prime powers 13 <= q <= 20000
TOTAL failures: 0   (running time 81 s)
