Hall–Petresco identity

Template:Short description In mathematics, the Hall–Petresco identity (sometimes misspelled Hall–Petrescu identity) is an identity holding in any group. It was introduced by Template:Harvs and Template:Harvs. 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

$x^m{y}^m=(xy{)}^m{c}_2^{(\genfrac{}{}{0ex}{}{m}{2})}{c}_3^{(\genfrac{}{}{0ex}{}{m}{3})}\cdots c_{m-1}^{(\genfrac{}{}{0ex}{}{m}{m-1})}{c}_m$

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

References

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