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On centrally nilpotent loops. (English) Zbl 1038.20052

Following R. Baer, the authors prove that \(H\) is a subnormal subgroup of depth at most \(n\) in a group \(G\) if and only if the same is true when \(G\) is replaced by \(\langle H,X\rangle\), for every finite \(X\subseteq G\). This lemma is then used to examine the problem when the nilpotency of \(M(Q)\) implies the central nilpotency of \(Q\), and vice versa, where \(Q\) is a loop and \(M(Q)\) is its multiplication group. The authors obtain a certain technical result, which makes it possible to address the case of commutative A-loops. If \(Q\) is such a loop, then \(Q\) is centrally nilpotent of class at most \(n\) if and only if \(M(Q)\) is nilpotent of class at most \(2n-1\).

MSC:

20N05 Loops, quasigroups
20E15 Chains and lattices of subgroups, subnormal subgroups
20F05 Generators, relations, and presentations of groups
20F18 Nilpotent groups
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