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.

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

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

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

• Baker–Campbell–Hausdorff formula
• Algebra of symbols

• Hall, Marshall (1959), The theory of groups, Macmillan, MR  0103215
• Hall, Philip (1934), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society, 36: 29–95, doi: 10.1112/plms/s2-36.1.29
• Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, pp. 90–93, ISBN  978-3-540-03825-2, MR  0224703, OCLC  527050
• Petresco, Julian (1954), "Sur les commutateurs", Mathematische Zeitschrift, 61 (1): 348–356, doi: 10.1007/BF01181351, MR  0066380