check_note.py: exact checks for OWR-12978-004
== A: digit identities
PASS A1 s + t = (s + t mod 3) + 3 carry(s, t) for s, t in {0,1,2}
PASS A2 carry is a 2-cocycle on Z/3 (identity holds in Z)
PASS A3 carry(t, t) + carry(2t mod 3, t) = t for t = 0, 1, 2
== B: group law of P1
PASS B1 [P1] cocycle identity c(x,y)+c(x+y,u) = c(y,u)+c(x,y+u) on all 81^3 triples
PASS B2 [P1] c(0, w) = c(v, 0) = 0
PASS B3 [P1] (0,0) is a two-sided identity
PASS B4 [P1] every element has a unique two-sided inverse
PASS B5 [P1] associativity on a sample of 400995 triples
== C: commutators, centre, classes of P1
PASS C1 [P1] [(v,0),(w,0)] = (0, beta(v,w)) for all v, w
PASS C2 [P1] beta(v, .) is onto F_9 for every v != 0
PASS C3 [P1] Z(P) = 0 x Z
PASS C4 [P1] the set of commutators is 0 x Z
PASS C5 [P1] 89 classes: 9 of size 1, 80 of size 9 (found 89, {1: 9, 9: 80})
PASS C6 [P1] every non-central class is the coset gZ (Camina property)
== D: cubes and orders in P1
PASS D1 [P1] (v,z)^3 = (0, phi(v)) for all 729 elements
PASS D2 [P1] element orders {1: 1, 3: 80, 9: 648}, exponent 9 (found {1: 1, 3: 80, 9: 648})
PASS D3 [P1] |P:P'| = 81, |P'| = 9, |mho_1(P)| = |{g^3}| = 9 (Lewis's data)
PASS D4 [P1] the projection P -> V is a homomorphism (all pairs)
== E: characters of P1
PASS E1 [P1] the 89 functions are class functions
PASS E2 [P1] linear characters multiplicative (11 characters x 146 x 729 pairs)
PASS E3 [P1] Ind_Z^P(lambda) = 9 chi_lambda for the 8 non-trivial lambda (brute force)
PASS E4 [P1] the 89 class functions are exactly orthonormal
PASS E5 [P1] degrees 81 x 1 and 8 x 9, sum of squares 729
PASS E6 [P1] fusion rules: psi_xi psi_eta = psi_(xi+eta), psi_xi chi_l = chi_l, chi_l chi_m = 9 chi_(l+m) (l+m != 0), chi_l chi_(-l) = sum of the 81 linear characters
== B: group law of P2
PASS B1 [P2] cocycle identity c(x,y)+c(x+y,u) = c(y,u)+c(x,y+u) on all 81^3 triples
PASS B2 [P2] c(0, w) = c(v, 0) = 0
PASS B3 [P2] (0,0) is a two-sided identity
PASS B4 [P2] every element has a unique two-sided inverse
PASS B5 [P2] associativity on a sample of 400995 triples
== C: commutators, centre, classes of P2
PASS C1 [P2] [(v,0),(w,0)] = (0, beta(v,w)) for all v, w
PASS C2 [P2] beta(v, .) is onto F_9 for every v != 0
PASS C3 [P2] Z(P) = 0 x Z
PASS C4 [P2] the set of commutators is 0 x Z
PASS C5 [P2] 89 classes: 9 of size 1, 80 of size 9 (found 89, {1: 9, 9: 80})
PASS C6 [P2] every non-central class is the coset gZ (Camina property)
== D: cubes and orders in P2
PASS D1 [P2] (v,z)^3 = (0, phi(v)) for all 729 elements
PASS D2 [P2] element orders {1: 1, 3: 80, 9: 648}, exponent 9 (found {1: 1, 3: 80, 9: 648})
PASS D3 [P2] |P:P'| = 81, |P'| = 9, |mho_1(P)| = |{g^3}| = 9 (Lewis's data)
PASS D4 [P2] the projection P -> V is a homomorphism (all pairs)
== E: characters of P2
PASS E1 [P2] the 89 functions are class functions
PASS E2 [P2] linear characters multiplicative (11 characters x 146 x 729 pairs)
PASS E3 [P2] Ind_Z^P(lambda) = 9 chi_lambda for the 8 non-trivial lambda (brute force)
PASS E4 [P2] the 89 class functions are exactly orthonormal
PASS E5 [P2] degrees 81 x 1 and 8 x 9, sum of squares 729
PASS E6 [P2] fusion rules: psi_xi psi_eta = psi_(xi+eta), psi_xi chi_l = chi_l, chi_l chi_m = 9 chi_(l+m) (l+m != 0), chi_l chi_(-l) = sum of the 81 linear characters
== F: the bijections tau (classes) and sigma (characters)
PASS F1 alpha is an involution of V and phi2(alpha(v)) = phi1(v) for all v
PASS F2 tau is a bijection of the 89 classes
PASS F3 tau preserves class sizes
PASS F4 sigma is a bijection of the 89 irreducible characters
PASS F5 sigma(chi)(tau C) = chi(C) for all chi, C (exact)
PASS F6 power maps n = 0..18 well defined on classes and tau(C^[n]) = tau(C)^[n]
== G: higher indicators
PASS G1 [P1] all nu_n (n = 1..18) are integers
PASS G2 [P1] nu_n depends only on n mod 9 (n <= 18)
PASS G3 [P1] class formula of Lemma 3.1 agrees with the brute force
     [P1] indicator sequences (nu_1, ..., nu_9):
         80 x (0, 0, 1, 0, 0, 1, 0, 0, 1)
          8 x (0, 0, 0, 0, 0, 0, 0, 0, 9)
          1 x (1, 1, 1, 1, 1, 1, 1, 1, 1)
     [P1] literal value set S (1 <= j <= 9): [0, 1, 9]
PASS G1 [P2] all nu_n (n = 1..18) are integers
PASS G2 [P2] nu_n depends only on n mod 9 (n <= 18)
PASS G3 [P2] class formula of Lemma 3.1 agrees with the brute force
     [P2] indicator sequences (nu_1, ..., nu_9):
         80 x (0, 0, 1, 0, 0, 1, 0, 0, 1)
          8 x (0, 0, 0, 0, 0, 0, 0, 0, 9)
          1 x (1, 1, 1, 1, 1, 1, 1, 1, 1)
     [P2] literal value set S (1 <= j <= 9): [0, 1, 9]
PASS G4 both multisets are 80 x (0,0,1,0,0,1,0,0,1), 8 x (0,...,0,9), 1 x (1,...,1)
PASS G5 nu_n(sigma chi) = nu_n(chi) for all 89 chi and n = 1..18
== H: Omega = {g : g^3 = 1}
     [P1] |Omega| = 81, subgroup: True, abelian: True
     [P2] |Omega| = 81, subgroup: True, abelian: False
PASS H1 Omega has order 81 and is a subgroup in both groups
PASS H2 Omega(P1) is abelian and Omega(P2) is not
PASS H3 ker phi1 is isotropic for beta; ker phi2 is not
== I: C^{Heis(3)} and C[(Z/3)^3]
     Heis(3): exponent 3, commutative: False, sequences nu_1..nu_9: {(1, 1, 1, 1, 1, 1, 1, 1, 1): 1, (0, 0, 1, 0, 0, 1, 0, 0, 1): 26}
     (Z/3)^3: exponent 3, commutative: True, sequences nu_1..nu_9: {(1, 1, 1, 1, 1, 1, 1, 1, 1): 1, (0, 0, 1, 0, 0, 1, 0, 0, 1): 26}
PASS I1 same indicator data 26 x (0,0,1,...), 1 x (1,...,1) and exponent 3
PASS I2 Heis(3) is non-abelian, (Z/3)^3 is abelian (Grothendieck rings Z[G] differ)
== J: Q8 (conditional remark on Question 7(a))
PASS J0 the multiplication of Q8 is associative
PASS J1 Q8 has exactly three subgroups of order 4, all containing -1 (found 3)
PASS J2 every subgroup K of order 4 meets only 2 cosets of {1,-1}: restriction H' -> C^K has rank 2 < 4
== K: control, D8 and Q8
PASS K1 8 nu_2 of the 2-dimensional character: D8 8, Q8 -8 (nu_2 = +1 and -1)
TOTAL failures: 0   (running time 9 s)
