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On connected transversals in PSL\((2,q)\). (English) Zbl 0744.20058

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.

MSC:

20N05 Loops, quasigroups
20D06 Simple groups: alternating groups and groups of Lie type

Citations:

Zbl 0706.20046
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