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On loops whose inner permutations commute. (English) Zbl 1101.20034
Let $$Q$$ be a loop, $$G=M(Q)$$ its multiplication group, and $$H=I(Q)$$ its inner mapping group. The paper studies certain aspects of $$G$$ under the assumption that $$H$$ is Abelian.
Denote by $$K$$ the normal closure of $$H$$ in $$G$$. Then $$K=G'H$$, for every loop (Proposition 3.1). If $$H$$ is Abelian, then $$Z(G)\cap K\neq 1$$, $$Z(K)\neq 1$$ and $$Z(K)H\neq H$$ (Proposition 3.6). In Section 4 one assumes, in addition, the existence of $$P\leq Z(G)\cap K$$ with $$PH\trianglelefteq K$$. This gives a number of consequences; for example that $$G'''=1$$ and that $$K$$ is nilpotent of class at most two. In the last section one proves, amongst others, that if $$Q$$ is nilpotent of class at least three, then every prime dividing $$|H|$$ divides $$|Q|$$.
The paper uses the language of $$H$$-connected transversals. That makes it look very formal. However, when a translation is made into the language of structural loop theory, the statements usually acquire a clear meaning. In a few cases one even discovers that rather obvious facts are being proved – for example Lemma 2.10 is a veiled form of saying that a loop is a group if and only if the left and the right translations commute.

##### MSC:
 20N05 Loops, quasigroups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D40 Products of subgroups of abstract finite groups
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