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On loops which have dihedral \(2\)-groups as inner mapping groups. (English) Zbl 0838.20080
If a loop has the property mentioned in the title then its multiplication group is solvable and \(Q\) is centrally nilpotent. It is known, that many properties of loops can be reduced to the properties of so called connected transversals in the multiplication group. First some properties of connected transversals are listened. Then it is proved that if a finite group \(G\) has a subgroup \(H\) such that \(H\) is a dihedral 2-group and there exists \(H\)-connected transversals \(A\) and \(B\) in \(G\), then \(G\) is solvable. By using a new argument on centralisers the author shows that also in the case of infinite \(G\) the above result is true. Assumed in addition that \(G\) is generated by \(A\cup B\), \(H\) is subnormal in \(G\). The main theorem about the central nilpotency of loops is an application of this result.

MSC:
20N05 Loops, quasigroups
20D15 Finite nilpotent groups, \(p\)-groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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