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On Bol loops of order 4k. (English) Zbl 0545.20059

The author establishes conditions on the automorphisms \(\alpha\), \(\beta\) of a group H such that the cartesian product \(G\times H\) of two groups is a Bol loop under the operation defined by \((a,x)(b,y)=(ab,x^{\beta(b)}y^{\alpha(a)}).\) The construction is a generalization of a construction due to K. H. Robinson [Aequationes Math. 22, 302-306 (1981; Zbl 0478.20044)]. By taking \(G=D_ 2\) and \(H=C_ k\), two non-isomorphic Bol loops of order 4k are constructed. For odd k these particular loops have previously appeared in the literature. For even k, one was given by Robinson and the other is new.
It should be noted that there is a plentiful supply of nonassociative Bol loops of order 4k for k even. The reviewer identified six, in isotopic triples [in Math. Proc. Camb. Philos. Soc. 89, 445-455 (1981; Zbl 0462.20056)]. Five are noted in the paper of Robinson already cited. Six may be constructed as direct products of a cyclic group of order \({1\over2}^ k \)and a Bol loop of order 8, and two may be constructed as direct products of a cyclic group of order 2 and a Bol loop of order 2k.
Reviewer: R.P.Burn

MSC:

20N05 Loops, quasigroups
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