On the regular power structure of $p$-groups and applications

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 $G$.