The Number of Finite Groups Whose Element Orders is Given
| dc.creator | Moghaddamfar, A. R. | |
| dc.creator | Shi, W. J. | |
| dc.date | 2005-09-22 | |
| dc.date | 2005-10-08 | |
| dc.date.accessioned | 2026-07-07T06:43:05Z | |
| dc.date.available | 2026-07-07T06:43:05Z | |
| dc.description | 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 $a^n-1$. | |
| dc.description | 17 pages | |
| dc.identifier | https://arxiv.org/abs/math/0509505 | |
| dc.identifier | http://arxiv.org/abs/math/0509505 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/102283 | |
| dc.subject | Group Theory | |
| dc.subject | 20D05 | |
| dc.title | The Number of Finite Groups Whose Element Orders is Given | |
| dc.type | text |