Commutators and Conjugates
Commutators and conjugates are group-theoretic notions with important applications in twisty puzzles. They are formal versions of the simple, intuitive idea of "do something to set up another task which does something useful, and undo the setup." Commutators can be used to generate algorithms that only modify specifics portion of a cube, and are intuitive derivable. Many puzzle solutions are heavily or fully based on commutators.
Given a group, a commutator is an element of the form ghg'h' (also denoted [g, h]), where g and h are elements of the group with inverses g' and h'. A conjugate of an element g is an element of the form hgh'. In cubing terminology, a commutator is an algorithm of the form A B A' B', and a conjugate is an algorithm of the form A B A', where A and B denote arbitrary algorithms and A', B' denote their respective inverses.
Notation
- [A, B] is a commonly used notation to represent the sequence A B A' B'.
- [A: B] is a well-accepted representation of the conjugate A B A'.
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 a commutator, and a colon or a semicolon a conjugate. The symbols are given order of precedence: colon, comma, semicolon. For example, [A; B, C: D] represents A ( B (C D C') B' (C D' C')) A'.
See also
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.
- 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.
- Build your own algorithms with commutators without using any notation.
- Youtube: YouTube tutorial by badmephisto on Commutators and their uses
- Speedsolving.com: Post from Speedsolving on Commutators
