The Collection Formula and Regular p-Groups

Abstract

For any elements x, y of a group G and for any positive integer n, we obtain the collection formula for the n-th power of the product xy when all commutators in G that have more than 3 occurrences of the element y are equal to 1. Suppose n = p is a prime number, and any commutator weighting more than p − 1 has an order 1 or p. In terms of the segment of the formula including only commutators of weight p with the exponents modulo p, we provide a necessary condition for the regularity of a finite p-group. This result generalizes known necessary conditions that were used in proving the non-regularity of some matrix p-groups.