On the number of edges of cyclic subgroup graphs of finite groups (Q2693998)

For a finite group \(G,\) denote by \(\eta(G)\) the number of pairs \((H_1,H_2),\) where \(H_2\) is a cyclic subgroup of \(G\) and \(H_1\) is a maximal subgroup of \(H_2\). Notice that \(\eta(G)\) is the number of edges of the cyclic subgroup graph of \(G\). The author proves that if \(G\) has order \(n\) and either \(G\) is nilpotent or \(n\) is odd, then \(\eta(G)\geq \eta(C_n),\) where \(C_n\) is the cyclic group of order \(n.\) He conjectures that this holds for arbitrary finite groups.