Applications of dimension subgroups to the power structure ofp-groups

Dimension subgroups in characteristicp are employed in the study of the power structure of finitep-groups. We show, e.g., that ifG is ap-group of classc (p odd) andk=⌜log _p ((c+1)/(p−1))⌝, then, for alli, any product ofp ^i+k th powers inG is ap ⁱ th power. This sharpens a previous result of A. Mann. Examples are constructed in order to show that our constantk is quite often the best possible, and in any case cannot be reduced by more than 1. Partially supported by MPI funds. This author is a member of GNSAGA-CNR. Partially supported by a Rothschild Fellowship.