# 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 commonly used to represent the conjugate A B A'.

## Nested conjugate and commutator

The sequences used in a commutator can be conjugates, thus giving commutators of conjugates. For example, [[R: U], D] = (R U R') D (R U' R') D'

Conversely, the sequence embedded in a conjugate can be commutator, thus giving a conjugate of commutators. For examples, [F: [U, R]] = F (U R U' R') F'