On the number of cyclic subgroups of a group

Lower Bounds on the Number of Cyclic Subgroups in Finite Non-Cyclic Nilpotent Groups

“…Proof. There are 5 groups of order 2 3 , and 14 groups of order 2 4 . We use Magma to obtain a full list of the isomorphism classes of groups in each case, and ask Magma for the total number of cyclic subgroups of each group in the list.…”

“…They showed that |G| ≤ 8r. Cocke and Jensen [4] proved that if G is not a 2-group then |G| ≤ 6r. Jafari and Madadi [9] proved that for any a divisor m of |G|, G has at least d(m) cyclic subgroups whose orders divide m. Garonzi and Lima [5] studied the function α(G)= c (G) |G| .…”