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.

The Hall-Witt identity is a three-variable identity among iterated group commutators. Passing to the leading terms of a filtered group turns it into the Jacobi identity for the associated graded Lie algebra.

The Hall-Petrescu formula expands $(xy{)}^{p^n}$ as $x^{p^n}{y}^{p^n}$ times ordered powers of higher commutators. In a p-valued group, the correction terms have controlled higher valuation and make the p-power map linear on the associated graded group.