Groups with finitely many non-normal subgroups (Q1118031)

In this paper all groups \(G\) are classified that have only finitely many non-normal subgroups. It is proved that a group \(G\) has this property if and only if it is either Abelian, or Hamiltonian, or finite, or of the form \(A\times B\), where \(A, B\) are as follows: \(B\) contains for some prime number \(p\), a central subgroup \(C\) that is isomorphic to the group of complex roots of unity of \(p\)-power order, such that \(B/C\) is a finite Abelian \(p\)-group, and \(A\) is a finite Abelian or Hamiltonian group of order not divisible by \(p\). It follows from this result that any infinite group that has non-normal subgroups at all has at least six of them, with equality occurring for precisely one group, up to isomorphism. As an auxiliary result, all infinite Abelian groups \(G\) are described that have an automorphism that fixes almost all but not all subgroups of \(G\).