The intersection of subgroups of finite p-groups

Abstract

For a finite p-group G and a positive integer k let Ik(G) denote the intersection of all subgroups of G of order pk. This paper classifies the finite p-groups G with \({{I}_k(G)\cong C_{p^{k-1}}}\) for primes p > 2. We also show that for any k, α ≥ 0 with 2(α + 1) ≤ k ≤ n−α the groups G of order pn with \({{I}_k(G)\cong C_{p^{k-\alpha}}}\) are exactly the groups of exponent pn-α.