# Article

Full entry | PDF   (0.2 MB)
Summary:
$G(3,m,n)$ 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.
References:
[1] F.R. Beyl and J. Tappe: Group Extensions. Presentations and the Schur Multiplicator. Lecture Notes in Mathematics 958, Springer-Verlag, Berlin, 1982. MR 0681287
[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 10.1112/blms/12.1.17 | MR 0565476
[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 0760656
[4] H. S. Coxeter: The binary polyhedral groups and other generalizations of the quaternion group. Duke Math. J. 7 (1940), 367–379. MR 0003409 | Zbl 0024.15002
[5] B. Huppert: Endliche Gruppen I. Springer, Berlin, 1967. MR 0224703 | Zbl 0217.07201
[6] G. Karpilovsky: The Schur Multiplier. Oxford University Press, Oxford, 1987. MR 1200015 | Zbl 0619.20001
[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 10.1016/0040-9383(65)90064-9 | MR 0179234 | Zbl 0146.04002
[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 1226568
[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 1338491
[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 10.1007/BF00052109 | MR 1397552
[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 0679156 | Zbl 0502.20003

Partner of