Vesanen, Ari On connected transversals in PSL\((2,q)\). (English) Zbl 0744.20058 Annales Academiæ Scientiarum Fennicæ, Series A, I. Mathematica, Dissertationes. 84. Helsinki, Oulu: Suomalainen Tiedeakatemia, Univ. of Oulu, Fac. of Science. 43 p. (1992). This dissertation deals with the question: which finite simple groups are isomorphic to the multiplication groups of loops or quasigroups. T. Kepka and M. Niemenmaa [J. Algebra 135, 112-122 (1990; Zbl 0706.20046)] proved that a group \(G\) is isomorphic to the multiplication group of a loop iff it has a suitable subgroup \(H\) with connected transversals which generate the group \(G\). They also showed that in a simple group, there can be connected transversals to maximal subgroups only. The author concentrates on the projective special linear groups \(PSL(2,q)\), where \(q\) is odd. The maximal subgroups of \(PSL(2,q)\) are known. Thus the investigation is partitioned in the following cases: (i) the solvable maximal subgroups of order \(q(q-1)/2\), (ii) the maximal subgroups of dihedral type, (iii) the maximal subgroups of the types \(A_ 4\), \(S_ 4\) and \(A_ 5\) and (iv) the remaining cases of the maximal subgroups.The main result is: the group \(PSL(2,q)\), where \(q>59\) is odd, is not isomorphic to the multiplication group of a quasigroup. Reviewer: E.Brožíková (Praha) Cited in 8 Documents MSC: 20N05 Loops, quasigroups 20D06 Simple groups: alternating groups and groups of Lie type Keywords:finite simple groups; multiplication groups; loops; quasigroups; connected transversals; projective special linear groups; maximal subgroups Citations:Zbl 0706.20046 PDFBibTeX XMLCite \textit{A. Vesanen}, On connected transversals in PSL\((2,q)\). Helsinki: Suomalainen Tiedeakatemia; Oulu: Univ. of Oulu, Fac. of Science (1992; Zbl 0744.20058)