In [4], Tȃrnȃuceanu described the finite groups G having exactly |G| − 1 cyclic subgroups. In [1], the authors used elementary methods to completely characterize those finite groups G having exactly |G| − ∆ cyclic subgroups for ∆ = 2, 3, 4 and 5. In this paper, we prove that for any ∆ > 0 if G has exactly |G| − ∆ cyclic subgroups, then |G| ≤ 8∆ and therefore the number of such G is finite. We then use the computer program GAP to find all G with exactly |G| − ∆ cyclic subgroups for ∆ = 1, . . . , 32.
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