The Number of Finite Groups Whose Element Orders is Given

The spectrum $\omega (G)$ of a finite group $G$ is the set of element orders of $G$. If $\mathrm{\Omega }$ is a non-empty subset of the set of natural numbers, $h(\mathrm{\Omega })$ stands for the number of isomorphism classes of finite groups $G$ with $\omega (G)=\mathrm{\Omega }$ and put $h(G)=h(\omega (G))$. We say that $G$ is recognizable (by spectrum $\omega (G)$) if $h(G)=1$. The group $G$ is almost recognizable (resp. nonrecognizable) if $1<h(G)<\mathrm{\infty }$ (resp. $h(G)=\mathrm{\infty }$). In the present paper, we focus our attention on the projective general linear groups $PGL(2,p^n)$, where $p=2^{\alpha }{3}^{\beta }+1$ is a prime, $\alpha \geq 0, \beta \geq 0$and$n\geq 1$, and we show that these groups cannot be almost recognizable, in other words $h(PGL(2,p^n))\in \{1,\mathrm{\infty }\}$. 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 $a^n-1$.