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On Efficient Presentations of the Groups PSL $(2, m)$ | ||
| International Journal of Group Theory | ||
| مقاله 1، دوره 11، شماره 3، آذر 2022، صفحه 131-150 اصل مقاله (645.69 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2021.128791.1696 | ||
| نویسنده | ||
| Orlin Stoytchev* | ||
| Department of Mathematics and Science, American University in Bulgaria, 1 Georgi Izmirliev Square, 2700 Blagoevgrad, Bulgaria | ||
| چکیده | ||
| 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 $J_2$. | ||
| کلیدواژهها | ||
| Von Dyck groups؛ Braid groups؛ Chevalley groups | ||
| مراجع | ||
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[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) 310–311. | ||
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