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Construction of finite loops of even order. (English) Zbl 0838.20079
Fong, Yuen (ed.) et al., Near-rings and near-fields. Proceedings of the conference, held in Fredericton, New Brunswick, Canada, July 18-24, 1993. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 336, 169-179 (1995).
The interest in \(K\)-loops, first introduced as the additive structures of H. Karzel’s near-domains [Abh. Math. Semin. Univ. Hamb. 32, 191-206 (1968; Zbl 0162.24101)], has grown considerably since it was shown by H. Wefelscheid that \(K\)-loops appear also as the structure of relativistic velocity composition studied by A. A. Ungar [Result. Math. 17, No. 1/2, 149-168 (1990; Zbl 0699.20055)]. Every \(K\)-loop is also a Bruck loop. In the paper under review, the author uses a method of A. Kreuzer and H. Wefelscheid [the author, ibid. 23, No. 3/4, 355-362 (1993; Zbl 0788.20036)] to construct various examples of Bol loops, Bruck loops, and \(K\)-loops. In particular, it is shown that for natural \(n\) and \(k\), non-isomorphic \(K\)-loops \((L,\oplus)\) of order \(8kn\) exist, having commutative subgroups \((G,\oplus)\) and \((H,\oplus)\) of order \(4n\) and \(2k\), respectively, with \(L= G\times H\).
For the entire collection see [Zbl 0824.00027].
Reviewer: R.Artzy (Haifa)

20N05 Loops, quasigroups