The number of subgroups of a finite group (II)

Abstract

Let A be a finite group, and let p be a prime. Suppose that ps is the highest power of p dividing , where is the commutator subgroup of A, and that the type with of a Sylow p-subgroup of satisfies either or and . Let denote the number of subgroups of index d in A. If and q is a positive integer such that , then is a multiple of pi and is a multiple of .