On the notion of gyrogroup

together with CnoFile{1619832}{} Einstein's relativistic velocity addition oplus in the c-ball bold R^3_c of the Euclidean 3-space bold R^3 is nonassociative. Hence, the Einstein groupoid (bold R^3_c,oplus) is not a group. Abstracting the key features of Einstein's groupoid (bold R^3_c,oplus) [see W. Krammer and H. K. Urbantke, Results Math. {bf 33} (1998), no.~3-4, 310--327; [msn] MR1619832 (99i:83003) [/msn]; see the preceding review], the notion of gyrogroup [A. A. Ungar, Found. Phys. {bf 27} (1997), no.~6, 881--951; [msn] MR1477047 (98k:83002) [/msn]] was introduced by the reviewer [Amer. J. Phys. {bf 59} (1991), no.~9, 824--834; [msn] MR1126776 (92g:83003) [/msn]] to describe a grouplike object that shares analogies with groups. The analogies stem from the relativistic effect called Thomas gyration (or, precession), and are emphasized by the prefix "gyro" that is extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. Furthermore, the reviewer introduced the gyrosemidirect product group as the product of a gyrogroup and a group that results in a group, in full analogy with the semidirect product group between two groups. The reviewer gave a brief history of the idea of a product of a nongroup groupoid and a group which results in a group, tracing it to Tits, Karzel and Kikkawa [A. A. Ungar, Aequationes Math. {bf 47} (1994), no.~2-3, 240--254; [msn] MR1268034 (95b:30076) [/msn]]. par However, the authors of the paper under review show that the idea should be traced to Sabinin's work, which has been rediscovered by Kikkawa. The authors justifiably claim that the reviewer did not give appropriate credit to Sabinin's pioneering work in this area. They describe the connections between gyrogroup theory and loop theory demonstrating that a gyrogroup is a Bol loop with Bruck identity, and that the reviewer's gyrosemidirect product is a construction that Sabinin has considered in a more general case since 1972 [P. O. Mikheev and L. V. Sabinin, in {it Quasigroups and loops: theory and applications}, 357--430, Heldermann, Berlin, 1990; [msn] MR1125818 [/msn]]. Finally, the authors remark that the discovery of gyrogroups in mathematical physics shows the vitality of quasigroup and loop theory.