When do the Fibonacci invertible classes modulo M form a subgroup?

In this paper, we look at the invertible classes modulo M representable as Fibonacci numbers and we ask when these classes, say FM, form a multiplicative group. We show that if M itself is a Fibonacci number, then M ≤ 8; if M is a Lucas number, then M ≤ 7. We also show that if x ≥ 3, the number of M ≤ x such that FM is a multiplicative subgroup is O(x/(log x)1/8).