On the number of subgroups of given type in a finite p-group

In §1 we study the p-groups G containing exactly p + 1 subgroups of order p p and exponent p. A number of counting theorems and results on subgroups of maximal class and p-groups with few subgroups of given type are also proved. Counting theorems play crucial role in the whole paper. This paper is a continuation of (Ber1, Ber3, Ber4, BJ2). We use the same notation, however, for the sake of convenience, we recall it in the following paragraph. In what follows, p is a prime, n, m, k, s, t are natural numbers, G is a finite p-group of order |G|, o(x) is the order of x ∈ G, n(G) = h x ∈ G | o(x) ≤ p n i , ∗(G) = h x ∈ G | o(x) = p n i and ℧n(G) = h x p n | x ∈ Gi . A p-group G is said to be absolutely regular if |G/℧1(G)| 2 and cl(G) = m − 1. As usually, G ' , �(G), Z(G) denote the derived subgroup, Frattini subgroup and center of G, respectively. Let i = {H 2 and n > 3 for p = 2, let Mpn = h a, b | a p n 1 = b p = 1, a b = a 1+p n 2 i . Let D2m, Q2m and SD2m be dihedral, generalized quaternion and semidihedral groups of order 2 m , and let Cpn, Epn be cyclic and elementary abelian groups