Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements

Abstract

Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups G whose set of numbers of subgroups of possible orders n(G) has exactly two elements. We show that if G is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then G has a normal Sylow subgroup of prime order and G is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with n(G) = {1, q + 1} for some prime q. In particular, G is supersolvable under this condition.