Group commutator ($[x,y]=x^{-1}{y}^{-1}xy$)

The group commutator measures the failure of two elements of a group to commute. Under the convention $[x,y]=x^{-1}{y}^{-1}xy$, it is the identity exactly when $xy=yx$. The commutator subgroup is generated by these elements. The alternative convention $xy{x}^{-1}{y}^{-1}$ generates the same commutator subgroup; individual formulas must specify their convention.

Use the group commutator convention $[x,y]=x^{-1}{y}^{-1}xy$ and write $u^v=v^{-1}uv$ for conjugation. The Hall-Witt identity isIt holds in every group. Substituting the definitions and cancelling adjacent inverse letters proves the formula directly; the three cyclic factors are all necessary for that cancellation.

Modulo the fourth term of the lower central series, the triple group commutators are central, so the outer conjugations disappear. Replacing an inverse input changes the corresponding leading group commutator by its inverse. In additive notation for the graded Lie algebra associated with the lower central series, the identity becomesthe Jacobi identity. This explains how a group-theoretic cancellation identity supplies the Lie-theoretic compatibility of leading commutators.

The Hall-Petrescu collection formula corrects the commutative rule $(xy{)}^n=x^n{y}^n$ by ordered products of higher group commutators. Its simplest useful form occurs in a two-step nilpotent group. With $[x,y]=x^{-1}{y}^{-1}xy$,Here $[y,x]$ is central. The relation $yx=xy[y,x]$ shows that moving the new $x$ past $n$ copies of $y$ adds $n$ to its exponent; mathematical induction and $\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)=\left(\genfrac{}{}{0ex}{}{n}{2}\right)+n$ prove the formula.

For an odd prime number $p$, the coefficient $\left(\genfrac{}{}{0ex}{}{p}{2}\right)$ is divisible by $p$. Thus if the commutator subgroup in this two-step setting has exponent dividing $p$, the $p$th power map satisfies $(xy{)}^p=x^p{y}^p$. For $p=2$ the correction generally remains. In higher nilpotency class, further commutators enter from deeper terms of the lower central series; their collection coefficients, rather than an assumption of commutativity, control power calculations in powerful p-groups and p-valued groups.