Group operation

A group operation is the associative binary operation of a group. It has an identity element, and every element has an inverse element.

An operation is associative when $(xy)z=x(yz)$ whenever the products are defined.

The group commutator measures the failure of two elements to commute. Under the convention $[x,y]=x^{-1}{y}^{-1}xy$, it is the identity exactly when $xy=yx$.

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.