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ON EFFICIENT PRESENTATIONS OF THE GROUPS PSL(2,m).
- Source :
-
International Journal of Group Theory . 2022, Vol. 11 Issue 3, p131-150. 20p. - Publication Year :
- 2022
-
Abstract
- We exhibit presentations of the Von Dyck groups D(2,3,m), m≥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≥6 we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for m≥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 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 SL(2,m) and PSL(2,m), Canadian J. Math., 24 (1972) 1129--1131]. The second applies to the cases m≡±2 (mod 3), m≡/ 11(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 J2. [ABSTRACT FROM AUTHOR]
- Subjects :
- *FINITE simple groups
*ARTIN algebras
*HYPERBOLIC groups
Subjects
Details
- Language :
- English
- ISSN :
- 22517650
- Volume :
- 11
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- International Journal of Group Theory
- Publication Type :
- Academic Journal
- Accession number :
- 159587010
- Full Text :
- https://doi.org/10.22108/IJGT.2021.128791.1696