On Efficient Presentations of the Groups PSL $(2, m)$

Document Type : Research Paper

Author

Department of Mathematics and Science, American University in Bulgaria, 1 Georgi Izmirliev Square, 2700 Blagoevgrad, Bulgaria

Abstract

We exhibit presentations of the Von Dyck groups $D(2, 3, m), \ m\ge 3$, in terms of two generators of order $m$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $m=3, 4, 5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $m\ge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for $m\ge 7$ when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain three-relator presentations of $\text{PSL}(2,m)$. We discover two general formulas presenting these as factors of $D(2, 3, m)$. The first one works for any odd $m$ and is essentially equivalent to the shortest known presentation of Sunday [J. Sunday, Presentations of the groups ${\rm SL}(2,\,m)$ and ${\rm PSL}(2,\,m)$, Canadian J. Math., 24 (1972) 1129--1131]. The second applies to the cases $m\equiv\pm 2\ (\text{mod}\ 3)$, $m\equiv \hskip -9pt/ \ 11(\text{mod}\ 30)$, and is substantively shorter. Additionally, by random search, we find many efficient presentations of finite simple Chevalley groups PSL($2,q$) as factors of $D(2, 3, m)$ where $m$ divides the order of the group. The only other simple group that we found in this way is the sporadic Janko group $J_2$.

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References

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International Journal of Group Theory
Volume 11, Issue 3 - Serial Number 3
September 2022
Pages 131-150

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History

  • Receive Date: 21 May 2021
  • Revise Date: 09 July 2021
  • Accept Date: 14 July 2021
  • Published Online: 01 September 2022

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