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

Document Type : Research Paper


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


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 $J_2$.


Main Subjects

[1] H. Behr and J. Mennicke, A presentation of the groups PSL(2, p), Canadian J. Math., 20 (1968) 1432–1438.
[2] C. M. Campbell, E. F. Robertson and P. D. Williams, Efficient presentations for finite simple groups and related
groups, Groups - Korea 1988, Lecture Notes in Math., 1398, Springer, Berlin, 1989 65–72.
[3] C. M. Campbell, G. Havas, C. Ramsay and E. F. Robertson, Nice efficient presentations for all small simple groups
and their covers, LMS J. Comput. Math., 7 (2004) 266–283.
[4] C. M. Campbell, G. Havas, C. Ramsay and E. F. Robertson, On the efficiency of the simple groups of order less
than a million and their covers, Experimental Math., 16 (2007) 347–358.
[5] C. M. Campbell and E. F. Robertson, A deficiency zero presentation for SL(2, p), Bull. London Math. Soc., 12
(1980) 17–20.
[6] M. D. E. Conder, Hurwitz groups: a brief survey, Bull. Amer. Math. Soc. (N.S.), 23 (1990) 359–370.
[7] H. S. M. Coxeter and W. 0. J. Moser, Generators and relations for discrete groups, Springer-Verlag, Berlin-GottingenHeidelberg, 1957.
[8] GAP - Groups, Algorithms, Programming-a System for Computational Discrete Algebra, https://www.gap-system.
[9] I. Hajdini and O. Stoytchev, The Fundamental Group of SO(n) Via Quotients of Braid Groups, Preprint (2016).
[10] F. Klein, On the order-seven transformation of elliptic functions, Translated from the German and with an introduction by Silvio Levy. Math. Sci. Res. Inst. Publ., 35, The eightfold way, Cambridge Univ. Press, Cambridge, 1999
[11] A. M. Macbeath, Hurwitz groups and surfaces, The eightfold way, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ.
Press, Cambridge, 1999 103–113.
[12] A. M. Macbeath, Generators of the linear fractional groups, 1969 Number Theory (Proc. Sympos. Pure Math., Vol.
XII, Houston, Tex., 1967), Amer. Math. Soc., Providence, 14–32.
[13] J. Mennicke, On Ihara’s modular group, Invent. Math., 4 (1967) 202–228.
[14] J. Schur, Untersuchungen uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J.
Reine Angew. Math., 132 (1907) 85–137.
[15] J. Sunday, Presentations of the groups SL(2, m) and PSL(2, m), Canadian J. Math., 24 (1972) 1129–1131.
[16] H. Zassenhaus, A presentation of the groups PSL(2, p) with three defining relations, Can. J. Math., 21 (1969)
Volume 11, Issue 3 - Serial Number 3
September 2022
Pages 131-150
  • Receive Date: 21 May 2021
  • Revise Date: 09 July 2021
  • Accept Date: 14 July 2021
  • Published Online: 01 September 2022