The Number of Finite Groups Whose Element Orders is Given

TL;DR

This paper investigates the spectrum of finite groups, especially ${PGL}(2,p^n)$, proving they are either uniquely determined by their spectrum or have infinitely many groups with the same spectrum, and provides computational tools for prime divisors.

Contribution

It establishes that certain projective general linear groups are either uniquely recognizable or nonrecognizable by their spectrum, and introduces a computer program for primitive prime divisor calculations.

Findings

${PGL}(2,p^n)$ groups are not almost recognizable, only either recognizable or nonrecognizable.

${PGL}(2,7)$ and ${PGL}(2,9)$ are nonrecognizable.

A computer program for primitive prime divisor detection is presented.

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, $\alpha \geq 0, \beta \geq 0$ and $n\geq 1$, and we show that these groups cannot be almost recognizable, in other words $h({PGL}(2,p^n))\in \{1, \infty\}$. 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 $a^n-1$.