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A construction of Bruck loops. (English) Zbl 0563.20053

This paper gives a construction for a left Bol loop (G,*) from an Abelian group \((G,+)\). The operation for the loop is defined by \(x*y=x+y+T(x,y,x+y),\) where \(T: G^ 3\to G\) is additive in each of its three variables, \(T(x,y,z)=T(x,z,y)\) and \(T(x,y,z)=0\) when x,y or \(z\in T(G^ 3)\). Elements inverse in \((G,+)\) are inverse in (G,*). Since \((- x)*(-y)=-(x*y),\) the loop is a Bruck loop. The author gives necessary and sufficient conditions on T for (G,*) to be (respectively) a Moufang loop or a group. In case \((G,+)=Z^ 3_ p\) with the usual vector addition, the mapping \(T(x,y,z)=(0,0,x_ 1y_ 2z_ 2)\) gives a proper Bol loop (G,*) of order \(p^ 3\).
Reviewer: R.P.Burn

MSC:

20N05 Loops, quasigroups
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