Hall–Petresco identity explained

In mathematics, the Hall–Petresco identity (sometimes misspelled Hall–Petrescu identity) is an identity holding in any group. It was introduced by and . 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 ^{my} ^{m=(xy)} ^{\binom{m} ^{2} { 2 }}c ^{\binom{m} _{3} { 3 }} … c ^{\binom{m} _{m -1} {m -1 }}c _{m} where each c_i is in the subgroup K_i of the descending central series of G.

See also

• Baker–Campbell–Hausdorff formula
• Algebra of symbols