On necessary conditions of regularity of Sylow $p$-subgroups of the group $GL_n(\mathbb{Z}_{p^m})$

In connect to B. Vehrfritz's problem 8.3 from Kourovka notebook we prooved that Sylow $p$-subgroup of group $GL_n(\mathbb{Z}_{p^m})$ when $n\leqslant (p-1)/2$ satisfies well-known necessary coditions of regularity.