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}.

**The Square**

Say I apply **R**, for **r**ight-hand **r**otation, to a square.

Clearly **R** has an inverse: a **l**eft 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****⁻****⁷** or **R²**. Which is just like a half-turn really. And notice **R² = L****²**.

Here’s what I can say **group**wise about **rotations of a square**:

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

** Noncommutative Manœuvres**

If you use **MS Paint** or other fine image editing tools, you know about **H**orizontal Flips and **V**ertical 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****⁰**. **V** is also cyclic and **V² = V****⁰**. 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?

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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Tags: abelian groups, Bryan Hayes, education, Felix Klein, group theory, isomorphism, math, mathematics, maths, matrices, Niels von Abel

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