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PART I t This is a special case of Bruck's form (6, Theorems 1, 11, 12) of Murdoch's result (18, Theorem 11) that every entropic quasigroup is isotopic to an abelian group. At the same time it is a special case of Bruck's construction (6, Theorem 3 and Theorem 7, Corollary 2) of certain inverse property quasigroups and t.s. quasigroups from groups. (At the end of the Corollary quoted, (53) would seem to be a misprint for (52).) Sade (21, No. 72; 22, p. 171) discusses the construction (5.1) anddoi:10.1017/s001309150000897x fatcat:nt5tpcwdlfe4lkuxjw3b66zlxu