Classification of Bol loops of order 18.

*(English)*Zbl 0545.20061The author establishes that there are exactly two nonassociative right Bol loops of order 18. The method used is to study the possible right multiplications. Each of the loops contains an elementary abelian subgroup of order 9, three elements of order 2 and six elements of order 6. Each loop is isomorphic to all its loop isotopes, but the two loops are not isomorphic to each other because their left nuclei are of orders 6 and 9 respectively. The author generalizes the constructions by defining a suitable multiplication on the set \(C_ p\times C_ p\times C_ 2\) for the first loop and on \(C_{2p}\times C_ p\) for the second loop. The definition for the second loop is incomplete (in the second (sic!) theorem 9) as right multiplication by \((X^ i,a^ j)\) is not defined for even i, nor for odd i if \(j<{1\over2}(p+1).\)

The reviewer has established that the representation of the first loop is a subset of the triangular group in \(GL(3,Z_ p)\) and the representation of the second loop is a subset of the group \(D_ p\times D_ p\times C_ p\), and he has also established a conjecture in this paper that all Bol loops of order \(2p^ 2\) are isomorphic to all their loop isotopes. The reviewer is grateful for this paper as he had previously overlooked the second of the two possible loops.

The reviewer has established that the representation of the first loop is a subset of the triangular group in \(GL(3,Z_ p)\) and the representation of the second loop is a subset of the group \(D_ p\times D_ p\times C_ p\), and he has also established a conjecture in this paper that all Bol loops of order \(2p^ 2\) are isomorphic to all their loop isotopes. The reviewer is grateful for this paper as he had previously overlooked the second of the two possible loops.

Reviewer: R.P.Burn

##### MSC:

20N05 | Loops, quasigroups |