A generalization of the theorems of Hall and Blackburn and their applications to nonregular $p$-groups

In this work we improve Philip Hall's estimate for the number of cyclic subgroups in a finite $p$-group. From our result it follows that if a $p$-group $G$ is not absolutely regular and not a group of maximal class, then 1) the number of solutions of the equation $x^p=1$ in $G$ is equal to $p^p + k(p-1)p^p$, where $k$ is a nonnegative integer; 2) if $n>1$, then the number of solutions of the equation $x^{p^n}=1$ in $G$ is divisible by $p^{n+p-1}$. This permits us to strengthen important theorems of Hall and Norman Blackburn on the existence of normal subgroups of prime exponent. The latter results in turn permit us to give a factorization of $p$-groups with absolutely regular Frattini subgroup. Another application is a theorem on the number of subgroups of maximal class in a $p$-group.