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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.

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.)
Full Text: DOI
[1] DOI: 10.1016/0021-8693(68)90050-1 · Zbl 0155.03901
[2] DOI: 10.1016/0021-8693(64)90017-1 · Zbl 0123.01502
[3] DOI: 10.1007/BF01388521 · Zbl 0564.20010
[4] DOI: 10.2307/1990147 · Zbl 0061.02201
[5] DOI: 10.1112/blms/24.4.343 · Zbl 0793.20064
[6] DOI: 10.1016/0021-8693(90)90176-O · Zbl 0718.17028
[7] DOI: 10.1016/0021-8693(90)90152-E · Zbl 0706.20046
[8] DOI: 10.1017/S0305004100067025 · Zbl 0622.20061
[9] Kepka, Bull. Austral. Math. Soc. 38 pp 171– (1988)
[10] Huppert, Endliche Gruppen I (1967) · Zbl 0217.07201
[11] Niemenmaa, Groups St.Andrews pp 396– (1991)
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