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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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