Revisiting the Leinster groups. (Q2438551)

Given a finite group \(G\), let \(\tau(G)\) be the number of normal subgroups of \(G\) and \(\sigma(G)\) be the sum of the orders of the normal subgroups of \(G\). A finite group \(G\) is a Leinster group if \(\sigma(G)=2\cdot|G|\). Clearly a finite cyclic group \(C_n\) is Leinster if and only if \(n\) is a perfect number, moreover the dihedral Leinster groups are in one-to-one correspondence with odd perfect numbers (one Leinster group of odd order is known: \((C_{127}\rtimes C_7)\times C_n\), \(n=3^4\cdot 11^2\cdot 19^2\cdot 113\)). In this paper the author classifies Leinster groups \(G\) with \(\tau(G)\leq 7\).