Finite p-groups whose non-normal subgroups have few orders

Abstract

Suppose that G is a nite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use p^{M(G)} and p^{m(G)} to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that M(G)\langle \mathrm{2}m(G)-\mathrm{1}: As a by-product, we also classify p-groups whose orders of non-normal subgroups are p^k and p^{k+\mathrm{1}}:

Key words： Finite p-groups; meta-hamiltonian p-groups; non-normal subgroups

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