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 = inverted right perpendicular log(p)((c + 1)/(p - 1)) inverted left perpendicular, 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.