The Number of Finite Groups Whose Element Orders is Given

The spectrum ω(G) of a finite group G is the set of element orders of G. If Ω is a non-empty subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups G with ω(G) = Ω and put h(G) = h(ω(G)). We say that G is recognizable (by spectrum ω(G)) if h(G) = 1. The group G is almost recognizable (resp. nonrecognizable) if 1 < h(G) < ∞ (resp. h(G) = ∞). In the present paper, we focus our attention on the projective general linear groups PGL(2, p), where p = 23 + 1 is a prime, α ≥ 0, β ≥ 0 and n ≥ 1, and we show that these groups cannot be almost recognizable, in other words h(PGL(2, p)) ∈ {1,∞}. It is also shown that the projective general linear groups PGL(2, 7) and PGL(2, 9) are nonrecognizable. In this paper a computer program has also been presented in order to find out the primitive prime divisors of a − 1. MSC 2000: 20D05