Applications of dimension subgroups to the power structure of p-groups

Dimension subgroups in characteristic p are employed in the study of the power structure of finite p-groups. We show, e.g., that if G is a p-group of class c (p odd) and k={top left corner}log _p ((c+1)/(p-1)){top right corner}, then, for all i, any product of p ^i+k th powers in G is a p ⁱ th power. This sharpens a previous result of A. Mann. Examples are constructed in order to show that our constant k is quite often the best possible, and in any case cannot be reduced by more than 1.