Exercise: Symmetries of the Square Index   <<   >>

  

i ←(⍳8),            1 2 3 0 7 4 5 6
i,←2 3 0 1 6 7 4 5, 3 0 1 2 5 6 7 4
i,←4 5 6 7 0 1 2 3, 5 6 7 4 3 0 1 2
i,←6 7 4 5 2 3 0 1, 7 4 5 6 1 2 3 0
i←8 8⍴i
D8← (⊂i)⌷1↓¨{(' '=⍵)⊂⍵}' ⊢ ⍒ ⍒⍒ ⍋⌽ ⌽ ⍋ ⍋⍒ ⍒⌽'
a.   Check that Cayley’s Theorem holds for D8 . 
r←, ⍳ ⊢
(r D8) ≡ r ∘.{⍺[⍵]}⍨ ↓r D8
b. Check that D8[i;0] composed with D8[0;j] is D8[i;j] 
p ← ?⍨ 97
(⍎¨D8,¨'p') ≡ ⍎¨(∘.,⍨0⌷D8),¨'p'

D8
∘.,⍨ 0⌷D8