k=1 |H_k| = 2 = 2^1
k=2 |H_k| = 8 = 2^3
PASS H_3 = Gamma_3 as sets and |Gamma_3| = 2^(3k-4)
k=3 |H_k| = 32 = 2^5   |Gamma_k| = 32   H_k == Gamma_k: True
PASS H_4 = Gamma_4 as sets and |Gamma_4| = 2^(3k-4)
k=4 |H_k| = 256 = 2^8   |Gamma_k| = 256   H_k == Gamma_k: True
PASS H_5 = Gamma_5 as sets and |Gamma_5| = 2^(3k-4)
k=5 |H_k| = 2048 = 2^11   |Gamma_k| = 2048   H_k == Gamma_k: True
PASS H_6 = Gamma_6 as sets and |Gamma_6| = 2^(3k-4)
k=6 |H_k| = 16384 = 2^14   |Gamma_k| = 16384   H_k == Gamma_k: True
PASS H_7 = Gamma_7 as sets and |Gamma_7| = 2^(3k-4)
k=7 |H_k| = 131072 = 2^17   |Gamma_k| = 131072   H_k == Gamma_k: True
PASS |H_3| = 32
PASS |H_4| = 256 = 8 |H_3|
PASS |H_(k+1)| = 8 |H_k| for 3 <= k < 7
PASS |PGL_2(Z/8)| = 384 = 6*8^2
PASS index of Gamma in PGL_2(Z_2) = 384/|Gamma_3| = 12
PASS k=3: delta Gamma delta^-1 = image of {g = 1 mod 2, det g = +-1} (32 elements, index 12)
PASS k=4: delta Gamma delta^-1 = image of {g = 1 mod 2, det g = +-1} (256 elements, index 12)
PASS k=5: delta Gamma delta^-1 = image of {g = 1 mod 2, det g = +-1} (2048 elements, index 12)
PASS k=6: delta Gamma delta^-1 = image of {g = 1 mod 2, det g = +-1} (16384 elements, index 12)
j=2: |K(j)/K(j+1)| = 8, classes reached by squares of K(j-1): 4
PASS negative control: for j=2 the squares of K(1) do NOT cover K(2)/K(3)
j=3: |K(j)/K(j+1)| = 8, classes reached by squares of K(j-1): 8
PASS Lemma 2.3(a): every class of K(3)/K(4) is a square of an element of K(2)
j=4: |K(j)/K(j+1)| = 8, classes reached by squares of K(j-1): 8
PASS Lemma 2.3(a): every class of K(4)/K(5) is a square of an element of K(3)
j=5: |K(j)/K(j+1)| = 8, classes reached by squares of K(j-1): 8
PASS Lemma 2.3(a): every class of K(5)/K(6) is a square of an element of K(4)
j=6: |K(j)/K(j+1)| = 8, classes reached by squares of K(j-1): 8
PASS Lemma 2.3(a): every class of K(6)/K(7) is a square of an element of K(5)
j=7: |K(j)/K(j+1)| = 8, classes reached by squares of K(j-1): 8
PASS Lemma 2.3(a): every class of K(7)/K(8) is a square of an element of K(6)
j=8: |K(j)/K(j+1)| = 8, classes reached by squares of K(j-1): 8
PASS Lemma 2.3(a): every class of K(8)/K(9) is a square of an element of K(7)
PASS commutators of K(i), K(j) lie in K(i+j) (3000 random pairs, i,j <= 5)

SUMMARY r2_4_groups: 0 failed
