Commutators and Conjugates

From Wiki

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 on a puzzle, and A', B' denote their respective inverses. 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.

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.