Commutators and Conjugates

Commutators and conjugates are a way of building sequences of moves. They refer to mathematical terms of group theory and have many applications in twisty puzzles, so that many puzzle solutions are heavily or fully based on commutators.

In intuitive terms, we can define them like this:

• conjugates: do something to set up another task which does something useful, and undo the setup (A B A').
• commutators: do something twice (the second time backwards) so that it gets almost completely canceled, allowing to modify specific portions of a cube (A B A' B').

Where A and B can by any sequence of moves.

Short notation

• [A, B] is a commonly used notation to represent the commutator A B A' B'.
• [A: B] is sometimes used to represent the conjugate A B A'.

Nested conjugate and commutator

The sequences used in a commutator can be conjugates, thus givin commutators of conjugates (for example (A B A') C (A B' A') C'). Conversely, the sequence embedded in a conjugate can be commutator, thus givin a conjugate of commutators (C (A B A' B') C').

Since commutators and conjugates are often nested together, Lucas Garron has proposed the following system for compact notation:

• brackets denote an entire algorithm, and within these
• the comma delimits both parts of a commutator like in a simple commutator
• the colon delimits both parts of a conjugate like in a simple conjugate
• the semicolon has the same meaning as the colon, but with lower precedence

The symbols are given order of precedence: colon, comma, semicolon.

For example:

• [B, C: D] = [B, [C: D]] = B (C D C') B' (C D' C')
• [B, C; D] = [[B, C]: D] = (B C B' C') D (C B C' B')
• [A; B, C: D] = A (B (C D C') B' (C D' C')) A'.

See also

• Commutator
• Conjugate
• Niklas
• Fewest Moves techniques
• BH
• Intuitive solving

External links

• RyanHeise.com has a page on commutators. Also notice all kinds of other useful sections on his website on the left. He has a lot of group theory related content, and other things that will really help you understand how algorithms work, and how you can create your own.
• Speedsolving.com: Post from Speedsolving on Commutators

Videos

• Build your own algorithms with commutators without using any notation.
• Commutators and Conjugates - The Ultimate Instructional Video - The title says it all, pay attention, take your time and make sure you have set some time aside before clicking play.
• Tutorial by badmephisto on Commutators and their uses