Any four things in order ABCD can be rearranged. You could do

  • ACBD — middle swap — 1324
  • DBCA — end swap — 4231
  • BCDA — rotation — 2341

for example.

Let’s say the four things are the four corners of a square. That’s valid, because ∃ a functor that maps    {rearranging the letters ABCD}    onto    {transformations of the square}.

permutation representation of the Klein Four group

The Square

Say I apply R, for right-hand rotation, to a square.

rotations of a square

Clearly R has an inverse: a left turn L (or anti-right turn R⁻¹ = L). Just follow the arrow in the opposite direction.

Moreover I can do arbitrary sequences of action R and R⁻¹, like say RRRRLLRLLRLRRLLR. And if I rearranged RLRL into RRLL, the outcome would be the same. (Think about that for a second, but yes it’s true.)

So I could do RRRRRRRRRLLLLLLL, let’s call it R⁹L⁷ for short, and since L is the opposite (“inverse”) of R, that’s just R⁹ or .  Which is just like a half-turn really. And notice R² = L².

Here’s what I can say groupwise about rotations of a square:

  • RL = LR. The group is commutative or “Abelian”.
  • R⁴ = R⁰. It’s a cyclic group of order 4.

Interaction between flips and rotations of the square.

Noncommutative Manœuvres

If you use MS Paint or other fine image editing tools, you know about Horizontal Flips and Vertical Flips. Interchange the leftmost pixel for the rightmost, the next in for its cousin across the board, etc.

Once you introduce either flip operation — call it H or V — and pair it with R and anti-R rotations, interesting things happen.

By itself, H is a cyclic group and H² = H.  V is also cyclic and V² = V. This implies that a flip is own inverse or V⁻¹ = V and H⁻¹ = H.

But what about VR? (Do the operations from left to right today.) What about VRV?  What about VVR?  I’ll draw the consequences and write the symbols but also play around for yourself and see what happens. Can you combine HVHHVVHV type operations to replicate RLRLRR type operations?

VR unachievable via R's alone


VRV = R⁻¹ = R³

Maybe it sounds hoity-toity when I use words like noncommutative or non-Abelian. I hope not.

Because basic facts like how objects turn over in your hands are what you learn as a toddler! Really, groups are just a mathematical way to talk about the way the world logically fits together; the way objects move in space, or … I’ll get to some human group mathematics another time.


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