These are my drawings of some simple groups: the “lightswitch group” `β€β`, the “hi-lo-off lightbulb” `β€β`, and the symmetric group `πβ=β€βΓβ€β`, which is how the letters `{A,B,C}` or the boyfriends `{Pankaj, Nadir, Ajay}` permute.

Edit: Several people have helpfully corrected me thatΒ `πββ β€βΓβ€β`. πβ needs to be theΒ twisted product ofΒ `β€β & β€β`, not a straight product.

$\dpi{300} \bg_white \large \begin{matrix} \mathrm{ABC} & \xrightarrow{\sigma_1_,_2} & \mathrm{BAC} & \xrightarrow{\sigma_2_,_3} & \mathrm{BCA} \\ &&&& \downarrow ^{\sigma_1_,_2} \\ &&&& \mathrm{CBA} \\ &&&& \downarrow ^{\sigma_2_,_3} \\ & & \mathrm{ABC} & \xleftarrow{\sigma_1_,_2} & \mathrm{CAB} \end{matrix}$

$\dpi{300} \bg_white \large \begin{matrix} \mathrm{ABC} & \overset{\sigma_1_,_2}\longleftrightarrow & \mathrm{BAC} & \overset{\sigma_2_,_3}\longleftrightarrow & \mathrm{BCA} \\ &&&& \updownarrow ^{\sigma_1_,_2} \\ &&&& \mathrm{CBA} \\ &&&& \updownarrow ^{\sigma_2_,_3} \\ & & \mathrm{ABC} & \overset{\sigma_1_,_2}\longleftrightarrow & \mathrm{CAB} \end{matrix}$

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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