Hall–Petresco identity
# Hall–Petresco identity

In mathematics, the Hall–Petresco identity (sometimes misspelled Hall–Petrescu identity) is an identity holding in any group. It was introduced by Hall (1934) and Petresco (1954). It can be proved using the commutator collecting process, and implies that p-groups of small class are regular.

## Statement

The Hall–Petresco identity states that if x and y are elements of a group G and m is a positive integer then

    $x^{m}y^{m}=(xy)^{m}c_{2}^{\binom {m}{2}}c_{3}^{\binom {m}{3}}\cdots c_{m-1}^{\binom {m}{m-1}}c_{m}$

where each $c_i$ is in the subgroup $K_i$ of the descending central series of G.

