A GENERALIZATION OF THE THEOREMS OF HALL AND BLACKBURN AND THEIR APPLICATIONS TO NONREGULAR $ p$-GROUPS

Abstract

In this work we improve Philip Hall's estimate for the number of cyclic subgroups in a finite -group. From our result it follows that if a -group is not absolutely regular and not a group of maximal class, then 1) the number of solutions of the equation in is equal to , where is a nonnegative integer; 2) if , then the number of solutions of the equation in is divisible by . 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 -groups with absolutely regular Frattini subgroup. Another application is a theorem on the number of subgroups of maximal class in a -group.