/Group Theory/Properties
Page 2 of 7

Group Properties

Discover how the Rubik's Cube group behaves: order, commutativity, and simplification

BasicsAlgorithms

How Does the Cube Group Behave?

Now that we know the Rubik's Cube forms a group, let's explore some important properties of this specific group:

  • Non-Commutativity: Does the order of moves matter? (Spoiler: YES!)
  • Element Order: How many times must you repeat a move sequence to get back to the start?
  • Move Cancellation: How can we simplify sequences by canceling opposite moves?

These properties help us understand how the cube group behaves and give us powerful tools for creating and analyzing algorithms.

1. Non-Commutativity (Order Matters!)

In most cases, doing move A then move B gives a different result than doing B then A. This means the cube group is non-commutative. Try different move combinations to see when they commute (rare!) and when they don't (common!).

Select two moves:and

R then U

U
L
F
R
D
B

U then R

U
L
F
R
D
B

13 stickers different (24.1% of cube)

✗ R • U ≠ U • R (Order matters!)

2. Order of Elements

Every move sequence has an 'order' - the number of times you must repeat it to return to the solved state. For example, R has order 4 (R⁴ = identity), while the famous Sexy Move has order 6!

Order of this sequence: 4

Repeat 4 times to return to solved state

U
L
F
R
D
B

At identity. Click "Apply Sequence" to begin.

3. Move Cancellation & Simplification

When analyzing or creating algorithms, we can simplify sequences by canceling opposite moves (like R R') and combining consecutive moves (like R R R → R'). This helps create more efficient solutions.

Original Sequence

R R'

2 moves

Simplified Sequence

(identity)

0 moves

Saved 2 moves (100.0% reduction)

Original: R R'

U
L
F
R
D
B

Simplified: (identity)

U
L
F
R
D
B

✓ Both sequences produce the same result, but the simplified version is more efficient!

Key Takeaways

  • Order matters! R U ≠ U R in most cases (non-commutativity)
  • Everything has finite order: Repeat any sequence enough times and you'll return to the start
  • Moves can cancel: R R' = identity, R R R = R'
  • Understanding these properties helps you create better algorithms and understand why certain sequences work