Geodesic
A new efficient presentation for $PSL(2,5)$ and the structure of the groups $G(3,m,n)$
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 1, pp. 67-74 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

$G(3,m,n)$ is the group presented by $\langle a,b\mid a^5=(ab)^2=b^{m+3}a^{-n}b^ma^{-n}=1\rangle $. In this paper, we study the structure of $G(3,m,n)$. We also give a new efficient presentation for the Projective Special Linear group $PSL(2,5)$ and in particular we prove that $PSL(2,5)$ is isomorphic to $G(3,m,n)$ under certain conditions.
$G(3,m,n)$ is the group presented by $\langle a,b\mid a^5=(ab)^2=b^{m+3}a^{-n}b^ma^{-n}=1\rangle $. In this paper, we study the structure of $G(3,m,n)$. We also give a new efficient presentation for the Projective Special Linear group $PSL(2,5)$ and in particular we prove that $PSL(2,5)$ is isomorphic to $G(3,m,n)$ under certain conditions.
Classification : 20D06, 20F05 
@article{CMJ_2000_50_1_a8,
     author = {Vatansever, Bilal and Gill, David M. and Eren, Nuran},
     title = {A new efficient presentation for $PSL(2,5)$ and the structure of the groups $G(3,m,n)$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {67--74},
     year = {2000},
     volume = {50},
     number = {1},
     mrnumber = {1745460},
     zbl = {1038.20021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2000_50_1_a8/}
}
TY  - JOUR
AU  - Vatansever, Bilal
AU  - Gill, David M.
AU  - Eren, Nuran
TI  - A new efficient presentation for $PSL(2,5)$ and the structure of the groups $G(3,m,n)$
JO  - Czechoslovak Mathematical Journal
PY  - 2000
SP  - 67
EP  - 74
VL  - 50
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2000_50_1_a8/
LA  - en
ID  - CMJ_2000_50_1_a8
ER  - 
%0 Journal Article
%A Vatansever, Bilal
%A Gill, David M.
%A Eren, Nuran
%T A new efficient presentation for $PSL(2,5)$ and the structure of the groups $G(3,m,n)$
%J Czechoslovak Mathematical Journal
%D 2000
%P 67-74
%V 50
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2000_50_1_a8/
%G en
%F CMJ_2000_50_1_a8
Vatansever, Bilal; Gill, David M.; Eren, Nuran. A new efficient presentation for $PSL(2,5)$ and the structure of the groups $G(3,m,n)$. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 1, pp. 67-74. http://geodesic.mathdoc.fr/item/CMJ_2000_50_1_a8/

[1] F.R. Beyl and J. Tappe: Group Extensions. Presentations and the Schur Multiplicator. Lecture Notes in Mathematics 958, Springer-Verlag, Berlin, 1982. | MR

[2] C. M. Campbell and E. F. Robertson: A deficiency zero presentation for ${\mathrm SL}(2,p)$. Bull. London Math. Soc. 12 (1980), 17–20. | DOI | MR

[3] J. J. Cannon: An introduction to the group language CAYLEY. Proc. Durham Symposium on Computational Group Theory, Academic Press, London, 1984, pp. 145–183. | MR

[4] H. S. Coxeter: The binary polyhedral groups and other generalizations of the quaternion group. Duke Math. J. 7 (1940), 367–379. | MR | Zbl

[5] B. Huppert: Endliche Gruppen I. Springer, Berlin, 1967. | MR | Zbl

[6] G. Karpilovsky: The Schur Multiplier. Oxford University Press, Oxford, 1987. | MR | Zbl

[7] I. Schur: Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 132 (1907), 85–137.

[8] R. G. Swan: Minimal resolutions for finite groups. Topology 4 (1965), 193–208. | DOI | MR | Zbl

[9] B. Vatansever: Certain Classes of Group Presentations. Ph.D. thesis University of St. Andrews, 1992.

[10] B. Vatansever and E. F. Robertson: A new efficient presentation for $PSL(2,13)$ and the structure of the groups $G(7,m)$. Doga-Tr. J. of Mathematics 17 (1993), 148–154. | MR

[11] B. Vatansever: A new efficient presentation for $PSL(2,37)$ and the structure of the groups $G(9,m)$. J. Inst. Math. Comput. Sci. Math. Ser. 7 (1994), 207–211. | MR

[12] B. Vatansever: New efficient presentations for $PSL(2,5)$ and $ SL(2,5)$; the structure of the groups $G(5,m)$ and $G(m,n,r)$. Acta Math. Hungar. 71 (1996), 205–210. | DOI | MR

[13] J. W. Wamsley: The Deficiency of Finite Groups. Ph.D. thesis University of Queensland, 1968.

[14] J. Wiegold: The Schur Multiplier. Groups (St. Andrews, 1981), C. M. Campbell and E. F. Robertson (eds.), LMS Lecture Notes, 71, Cambridge University Press, 1982, pp. 137–154. | MR | Zbl