One necessary condition for the regularity of a $p$-group and its application to Wehrfritz's problem

We obtain a necessary condition for the regularity of a $p$-group in terms of segments of P. Hall's collection formula. For any prime number $p$ such that $(p+2)/3$ is an integer, we prove that a Sylow $p$-subgroup of the group $GL_n(\mathbb{Z}_{p ^ m})$ is not regular if $n \geqslant (p+2)/3$ and $m \geqslant 3.$ We also list all regular Sylow $p$-subgroups of the Chevalley group of type $G_2$ over the ring $\mathbb{Z}_{p^m}.$