The Number of Finite Groups Whose Element Orders is Given

Abstract

The spectrum $\omega (G)$ of a finite group $G$ is the set of element orders of $G$. If $\Omega$ is a non-empty subset of the set of natural numbers, $h(\Omega )$ stands for the number of isomorphism classes of finite groups $G$ with $\omega (G)=\Omega$ 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 $1<h(G)<\infty$ (resp. $h(G)=\infty$). In the present paper, we focus our attention on the projective general linear groups $PGL(2,p^n)$, where $p=2^{\alpha }{3}^{\beta }+1$ is a prime, $\alpha \ge 0,\beta \ge 0$ and $n\ge 1$, 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 $a^n-1$.

17 pages