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

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

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

References

• 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