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In 1932, P.Hall proved the formula (known as Hall’s collection formula) in [1] that makes it possible to investigate interdependently the power and commutator structures of p−groups. The following statement was proved by P.Hall in the article on the theory of p−groups [1]. For any two elements x and y of any group G let the formally distinct complex commutators R1, R2, . . . of x and y be arranged in order of increasing weights (the order among the commutators of the same weight is arbitrary). Then for any natural n the following formula holds:

On Divisibility of Some Sums of Binomial Coefficients Arising From Collection Formulas