Bicrossproduct Hopf quasigroups

Abstract

We recall the notion of Hopf quasigroups introduced previously. We construct a bicrossproduct Hopf quasigroup $kM\bicross k(G)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_O$ as transversal in an order 128 group $X$ with subgroup $Z_2^3$ and hence obtain a Hopf quasigroup $kG_O\lcocross k(Z_2^3)$ as a particular case of our construction.