  In this paper, we study some variations of the well-known It\^{o}-Michler
theorem for $p$-Brauer characters using various inequalities involving the
$p$-Brauer character degrees of finite groups. Several new criteria for the
existence of a normal Sylow $p$-subgroup of finite groups are obtained using
the $p$-parts and $p'$-parts of the $p$-Brauer character degrees.
