The Number of Finite Groups whose Element Orders is Given

A. R. Moghaddamfar and W. J. Shi

Abstract: For any group $G$, $\pi_e(G)$ denotes the set of orders of its elements. If $\Omega$ is a non-empty 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

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