# Conjugate

A **conjugate** is a sequence of moves that consists in doing a sequence A, then a sequence B, and finally the inverse of the sequence A. In other words, a conjugate is an algorithm like A B A', where A and B can be any sequence of moves. In intuitive terms, it means: do something to set up another task which does something useful, and undo the setup.

## Contents

## Mathematical definition

Given a group, a **conjugate** is an element of the form *ghg'*, where *g* and *h* are elements of the group with inverses *g'* and *h'*.

Cube notation is very close : A B A'

It is sometimes written as [A: B] = A B A'

## Effect

In A B A', the sequence A is the setup move or sequence and A' puts the cube in its original state. In between, there is an embedded sequence B. Alone, this sequence would affect some set of locations **K** on the cube. Because of the conjugation, the affected pieces will be **K**A', which means that the locations will be transformed by the inverse of A.

Indeed the sequence B will be applied on a cube which space has been modified by A. The sequence B is supposed to change some pieces in the locations **K** and A consists in bringing the piece you want to change to the location **K**. Once you've done the sequence B, you bring those pieces back with A'.

### Trivial case

If the set of locations **K** is not modified by A, then the conjugation does not change anything, and A B A' = B.

### General case

Pieces can be affected by the following subsequence of ABA':

- by B only: this part is outside of the conjugation
- by AB: this is the first semi-conjugated part
- by BA': this is the second semi-conjugated part
- by ABA': this is the conjugated part

Let's call **J** the set of pieces modified by A, and **N** the intersection of **J** and **K**.

The initial location of the outside part is:
**O** = ((**K** \ **N**)B \ **N**)B'

The initial location of the first semi-conjugated part is:
**H**_{1} = (**N**B \ **N**)B'A'

The initial location of the second semi-conjugated part is:
**H**_{2} = **N**B' \ **N**

The initial location of the conjugated part is:
**C** = (**N**B inter **N**)B'A'

The slot location is:
**N**A'

### Usage

The conjugation is often used to provide a setup for some transformation. The conjugation ABA' is very similar to B, except that B is applied on other pieces as it would be without conjugation. For example, if a sequence exchange some pieces, or rotate some pieces, you can choose to exchange other pieces or rotate other pieces :

[F': [M2D2M2, U]]

where [M2D2M2, U] exchange the four top edges, so applying F' exchange the front-right edge instead of the front-top edge.

The intersection is **N** = front-top edge

The slot is **N**A' = **N**F = front-right edge

The conjugated part is:

**C** = (**N**B inter **N**)B'A' = empty

Surprisingly, in this example, the conjugated part is empty. It means that no individual piece is actually following the whole conjugation. F' and [M2D2M2, U] are only connected so that some pieces are transfered through the intersection **N**. The transfer occur between both semi-conjugated parts.

In this simple case, we do not really distinguish the semi-conjugated parts. But it is possible to take this difference into account, and also to use a conjugated part.

#### Semi-conjugated parts

Semi-conjugated parts can be used to move pieces inside or outside a slot. This is widely used to insert or extract a pair of pieces in F2L or to insert or extract corners or edges individually.

**H**_{1} is extracted from the slot and **H**_{2} is inserted into the slot.

Example: RU'R'

**N** = top-right edge, top-right-back and top-right-front corners

Slot location : **N**A' = **N**R' = top-right-front corner, front-right edge and front-right-bottom corner

**H**_{1} = (**N**U' \ **N**)UR' = top-right-front corner and front-right edge

These pieces are extracted from the slot and replace the top-back edge and top-back-left corner.

**H**_{2} = **N**U \ **N** = top-front-right corner and top-front edge

These pieces are inserted into the slot, replacing front-right edge, and front-right-bottom corner.

Example: RUR'

**N** = top-right edge, top-right-back and top-right-front corners

Slot location : **N**A' = **N**R' = top-right-front corner, front-right edge and front-right-bottom corner

**H**_{1} = (**N**U \ **N**)U'R' = front-right edge and front-right-bottom corner

These pieces are extracted from the slot and replace the top-front edge and top-front-left corner.

**H**_{2} = **N**U' \ **N** = top-back-left corner and top-back edge

These pieces are inserted into the slot, replacing front-right edge, and front-right-top corner.

#### Conjugated part

The conjugated part is used to move pieces inside the slot (the pieces do not enter nor leave the slot):

RUR'

where the intersection is **N** = top-right edge, top-right-front corner and top-right-back corner

and the slot is **N**A' = **N**R' = front-right edge, front-right-top corner and front-right-bottom corner

The conjugated part is :

**C** = (**N**B inter **N**)B'A' = top-right-front corner

This piece is moved to the front-right-bottom corner and rotated so that the right facelet is moved to the bottom facelet.

### Link with commutators

The outside part **O** is only affected by B', so it can be put back into place by adding B' to the sequence, giving ABA'B', which is the commutator of A and B.

The first semi-conjugated part **H**_{1}, which was affected by AB, is then only affected by A, becoming the container of pieces P_{A}.

The second semi-conjugated part **H**_{2} is split into two parts:

- one part which is affected by B', so affected by BA'B', and is then a conjugated part of the commutator
- one part which is not affected by B', so affected only by BA', becoming the container of pieces P
_{B}.

The conjugated part is split into two parts:

- one part which is affected by B', becoming the quirk part
- one part which is not, becoming a conjugated part of the commutator

Pieces that were not affected by the conjugate, but that are affected by the last B' move, are the P_{N} pieces of the corresponding commutator.