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On connected transversals to abelian subgroups. (English) Zbl 0799.20020
If \(G\) is a group, \(H\geq G\), \(A\) and \(B\) are two left transversals to \(H\) in \(G\), then we say that \(A\) and \(B\) are \(H\)-connected if \(a^{-1} b^{-1}\) \(ab\in H\) for every \(a\in A\) and \(b\in B\). This concept was introduced by the authors. They prove that if \(H\) is a finite abelian subgroup in \(G\) with two \(H\)-connected transversals \(A\) and \(B\), then \(G\) is solvable. If, in addition \(G =\langle A,B\rangle\) and the core of \(H\) in \(G\) is trivial, then \(Z(G)\neq 1\). They also investigate some special cases where \(H\) is very close to a cyclic group.
Finally the results are applied to loop theory and it is shown that if the inner mapping group of a finite loop is abelian, then the loop is centrally nilpotent.

MSC:
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20N05 Loops, quasigroups
20E07 Subgroup theorems; subgroup growth
20D25 Special subgroups (Frattini, Fitting, etc.)
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References:
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