Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 47, No. 2, pp. 463478 (2006) 

The Number of Finite Groups whose Element Orders is GivenA. R. Moghaddamfar and W. J. ShiDepartment of Mathematics, Faculty of Science, Imam Ali University, P. O. Box 13145689, Tehran, Iran; email: moghadam@iust.ac.ir; School of Mathematics, Soochow University, Suzhou 215006, People's Republic of China; email: wjshi@suda.edu.cnAbstract: For any group $G$, $\pi_e(G)$ denotes the set of orders of its elements. If $\Omega$ is a nonempty subset of $\mathbb{N}$, $h(\Omega)$ stands for the number of isomorphism classes of finite groups $G$ such that $\pi_e(G)=\Omega $. We put $h(G)=h(\pi_e(G))$. In this paper we show that $h(PGL(2,p^n))=1$ or $\infty$, where $p=2^\alpha 3^\beta+1$ is a prime, $\alpha \geq 0, \beta \geq 0$ and $n\geq 1$. In particular, we show that $h(PGL(2,7))=h(PGL(2,3^2))=\infty$. {\bf Editorial remark:} Due to a mixup of electronic files, the published version of the paper is not the final version submitted to the journal by the authors. The current second version represents the final version. It is available electronically only. The obsoleted first version, which appeared in print, is electronically archived <a href=b47h2shi.pdf>here</a>. Keywords: element order, prime graph, projective special linear group Classification (MSC2000): 20D05 Full text of the article:
Electronic version published on: 19 Jan 2007. This page was last modified: 5 Nov 2009.
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