Commutators

Definition 5.10.1   If $g, h$ are two elements of a group $G$ then we call the element

$$[g,h] = g*h*g^{-1}*h^{-1}$$

then commutator of $g, h$.

Not that $[g,h]=1$ if and only if $g, h$ commute. Thus the commutator may be regarded as a rough measurement of the lack of commutativity.

Definition 5.10.2 (Singmaster [Si])   Let $G$ be the permutation group generated by the permutations $R, L, U, D, F, B$ regarded as permutations in $S_{54}$. The Y commutator is the element

$$[F,R^{-1}] = F*R^{-1}*F^{-1}*R .$$

The Z commutator is the element

$$[F,R] = F*R*F^{-1}*R^{-1} .$$

Example 5.10.3   If $x,y$ are basic moves of the Rubik's cube associated to faces which share an edge then (a) $[x,y]^2$ permutes exactly 3 edges and does not permute any corners, (b) $[x,y]^3$ permutes exactly 2 pairs of corners and does not permute any edges.

Definition 5.10.4   Let $G$ be any group. The group $G'$ generated by all the commutators

$$\{ [g,h]\ \vert\ g, h \ {\rm belong\ to}\ G \}$$

This is called the commutator subgroup of G.

This group may be regarded as a rough measurement of the lack of commutativity of the group G.

Remark 5.10.5   We will see later that the group generated by the basic moves of the Rubik's cube - $R, L, U, D, F, B$ - has a relatively large commutator subgroup. In other words, roughly speaking ``most'' moves of the Rubik's cube can be generated by commutators such as the Y commutator or the Z commutator.

Exercise 5.10.6 (a)   Find the orders of the Y commutator and the Z commutator. (b) Find the order of $[R,[F,U]]$. (c) Why are these names used? (Think about the shapes of the squares affected on the Rubik's cube.)

Exercise 5.10.7   Let $G=S_3$, the symmetric group on 3 letters. Compute the commutators

$$[s_1,s_2],\ \ \ [s_2,s_1].$$

Exercise 5.10.8   Let $R, U$ be as in the notation for the Rubik's cube moves introduced in the previous chapter. Determine the order of the move $[R,U]$. (Ans: 6)