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
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
, and implies that ''p''-groups of small class are
regular
Regular may refer to:
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings
Other uses
* Regular character, ...
.
Statement
The Hall–Petresco identity states that if ''x'' and ''y'' are elements of a group ''G'' and ''m'' is a positive integer then
:
where each ''c''
''i'' is in the subgroup ''K''
''i'' of the descending central series of ''G''.
See also
*
Baker–Campbell–Hausdorff formula
In mathematics, the Baker–Campbell–Hausdorff formula gives the value of Z that solves the equation
e^X e^Y = e^Z
for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultima ...
*
Algebra of symbols
References
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{{DEFAULTSORT:Hall-Petresco identity
P-groups
Combinatorial group theory