On Strongly Quasi-regular P-groups

Let G be a finite p-group and s be a positive integer. G is called ps-quasi-regular if ?ab?^ps=1??? a^Psb^Ps=1 for any a,b ?G. In particular, if s=1, G is called quasi-regular. G is called strongly quasi-regular if G is pt-quasi-regular for every positive integer t. In this paper, it is proved that quasi-regular p-groups whose nilpotent class is less than 4 are p²-quasi-regular. In addition, it is also proved that quasi-regular A_t?t? 3? groups, Ct?t?3?groups and metahamiltonian p-groups are strongly quasi-regular.