Commutators and Conjugates
Introduction
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. Commutators are usually 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.
Commutators are related to, and often used in conjunction with [[Conjugate|Conjugates.
Commutator and Conjugate 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'.
Links
- Post from Speedsolving on Commutators
- YouTube tutorial by badmephisto on Commutators and their uses
- 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.
