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.
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 KA', 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: H1 = (NB \ N)B'A'
