The Number of Finite Groups Whose Element Orders is Given

Abstract

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 $1<h(G)<\infty$ (resp. $h(G)=\infty$). In the present paper, we focus our attention on the projective general linear groups ${PGL}(2,p^n)$, where $p=2^α3^β+1$ is a prime, $α\geq 0, β\geq 0$ and $n\geq 1$, and we show that these groups cannot be almost recognizable, in other words $h({PGL}(2,p^n))\in \{1, \infty\}$. 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 $a^n-1$.