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12. Cayley’s Theorem

r←,⍳⊢ ⋄ (r G) ≡ r ∘.{⍺[⍵]}⍨ ↓r G

The finite case of Cayley’s theorem [49a]: every group G is isomorphic to a subgroup of the group of permutations. For example:

   ⎕←T←2 2∘⍴¨(4⍴2)∘⊤¨9 6 7 11 13 14
┌───┬───┬───┬───┬───┬───┐
│1 0│0 1│0 1│1 0│1 1│1 1│
│0 1│1 0│1 1│1 1│0 1│1 0│
└───┴───┴───┴───┴───┴───┘
   ⎕←G←∘.(≠.∧)⍨T
┌───┬───┬───┬───┬───┬───┐
│1 0│0 1│0 1│1 0│1 1│1 1│
│0 1│1 0│1 1│1 1│0 1│1 0│
├───┼───┼───┼───┼───┼───┤
│0 1│1 0│1 1│1 1│0 1│1 0│
│1 0│0 1│0 1│1 0│1 1│1 1│
├───┼───┼───┼───┼───┼───┤
│0 1│1 0│1 1│1 1│0 1│1 0│
│1 1│1 1│1 0│0 1│1 0│0 1│
├───┼───┼───┼───┼───┼───┤
│1 0│0 1│0 1│1 0│1 1│1 1│
│1 1│1 1│1 0│0 1│1 0│0 1│
├───┼───┼───┼───┼───┼───┤
│1 1│1 1│1 0│0 1│1 0│0 1│
│0 1│1 0│1 1│1 1│0 1│1 0│
├───┼───┼───┼───┼───┼───┤
│1 1│1 1│1 0│0 1│1 0│0 1│
│1 0│0 1│0 1│1 0│1 1│1 1│
└───┴───┴───┴───┴───┴───┘

T are the non-singular 2-by-2 boolean matrices, and G is its group table under boolean matrix multiplication. The function r←,⍳⊢ relabels a group table according to its index in its ravel (cf. x⍳x); such relabelling, like displaying in a different font, does not affect group-theoretic properties.

   r ← , ⍳ ⊢
   r G
0 1 2 3 4 5
1 0 4 5 2 3
2 3 5 4 1 0
3 2 1 0 5 4
4 5 3 2 0 1
5 4 0 1 3 2

The rows of r G are permutations, and the relabeled version of its group table as permutations is the same as the relabelling of the original group:

   (r G) ≡ r ∘.{⍺[⍵]}⍨ ↓r G
1