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These are my drawings of some simple groups: the “lightswitch group” β„€β‚‚, the “hi-lo-off lightbulb” ℀₃, and the symmetric group π•Šβ‚ƒ=℀₂×℀₃, which is how the letters {A,B,C} or the boyfriends {Pankaj, Nadir, Ajay} permute.

Edit: Several people have helpfully corrected me thatΒ π•Šβ‚ƒβ‰ β„€β‚‚Γ—β„€β‚ƒ. π•Šβ‚ƒ needs to be theΒ twisted product ofΒ β„€β‚‚ & ℀₃, not a straight product.

Of course "swap 1 & 2" is its own inverse. So this "operation A, operation B, operation A, operation B, ..." switching is the *ONLY* alternative available to me to "progress further in the maze". It's then striking that the "braided" "alternating colours" of "operation A, operation B, operation A, operation B, ..." return me to the start in 5 moves. It works just as well for other pairs of swapping operations; you can draw it yourself. This little gem is hiding right in plain sight. An 8-year-old could have found it!

GLOSSARY

  • unary — takes one argument — like “square x”
  • binary — takes two arguments — like “x times y”
  • arity — how many arguments a function takes
  • function — theΒ target needs to be unique (per source) but not the other way around
  • inverse — when you can undo something
  • exponents — superscripts representing how many times you apply a verb







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    I’m taking two classes on this stuff next term. It looks super cool.
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  12. trismegistousse said: thanks
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