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On central nilpotency in finite loops with nilpotent inner mapping groups. (English) Zbl 1192.20057
Group-theoretic results are used to investigate further properties of the inner mapping group \(I(Q)\) of a finite loop \(Q\) which guarantee the central nilpotency of \(Q\) (it is known that such properties are, e.g., \(I(Q)\) being an Abelian group or a dihedral 2-group). Let \(H\) be a subgroup of a finite group \(G\), \(A\) and \(B\) be \(H\)-connected transversals in \(G\) and \(G=\langle A,B\rangle\). The authors prove that \(H\) is subnormal in \(G\) provided that either (i) \(H\cong D\times E\), where \(D\) is a dihedral group of order 8 and \(E\) is Abelian (Theorem 3.2), or \([G:H]=2^m\) and (ii) \(H\) is nilpotent and \(q\geq m\) for each odd prime \(q\) such that the Sylow \(q\)-subgroup of \(H\) is nonabelian (Theorem 3.4). These results are then used to show that a finite loop \(Q\) is centrally nilpotent if \(I(Q)=H\) satisfies the condition (i) or (ii) and \(|Q|=2^m\) in the latter case.
MSC:
20N05 Loops, quasigroups
20D15 Finite nilpotent groups, \(p\)-groups
20D35 Subnormal subgroups of abstract finite groups
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