  In this paper, we give elementary proofs of the Restricted Burnside Problem
and the Hughes Conjecture for finite $p$-groups with Hall's regular power
structure property. Moreover, in this setting we determine an explicit bound on
the order of a finite $d$-generator $p$-group of fixed exponent. Further
applications of $p$-groups with regular power structure are presented. For
example, we give a short new proof of an important property of powerful
$p$-groups; namely, that the minimal number of generators of a subgroup of such
a group $G$ is at most the number needed to generate $G$.
