Hall–Petresco identity
From HandWiki
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
- [math]\displaystyle{ x^my^m=(xy)^mc_2^{\binom{m}{2}}c_3^{\binom{m}{3}}\cdots c_{m-1}^{\binom{m}{m-1}}c_m }[/math]
where each ci is in the subgroup Ki of the descending central series of G.
See also
References
- Hall, Marshall (1959), The theory of groups, Macmillan
- 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) (in German), Endliche Gruppen, Berlin, New York: Springer-Verlag, pp. 90–93, ISBN 978-3-540-03825-2, OCLC 527050
- Petresco, Julian (1954), "Sur les commutateurs", Mathematische Zeitschrift 61 (1): 348–356, doi:10.1007/BF01181351
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