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A class of non-Moufang Bol loops isomorphic to all their loop isotopes. (English) Zbl 0583.20055
If G is a group, and on the cartesian product \(G\times G\) an operation \(\otimes\) is defined by \((a,b)\otimes (c,d)=(ac,b^ cd)\) where \(b^ c=cbc^{-1}\), then (G\(\times G,\otimes)\) is a left Bol loop. In case G is a nonabelian group of order pq where p and q are prime numbers and \(p| q-1\), the left Bol loop (G\(\times G,\otimes)\) contains a subloop of order \(pq^ 2\) which is not a Moufang loop and which is isomorphic to all its loop isotopes.
An isomorphic family of Bol loops, which are necessarily isomorphic to all their loop isotopes, may also be defined on the set \(Z_ p\times Z_ q\times Z_ q\) by means of the operation \(\cdot\) given by \((r,s,t)\cdot (x,y,z)=(r+s,sh^ x+y,yh^ r-y+t+z)\) where \(h\in Z_ q\), \(h\neq 1\), and \(h^ p=1\), and indices have their ordinary meaning in \(Z_ q\). The left nuclei of these loops have order \(q^ 2\) and the middle and right nuclei have order q. In case p is an odd prime, it would be interesting to know whether these loops are Bruck loops.
Reviewer: R.P.Burn

MSC:
20N05 Loops, quasigroups
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References:
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